A quadratic polynomial, when equated to zero, becomes a quadratic equation. The values of x satisfying the equation are called the roots of the quadratic equation.

**General from:** ax^{2} + bx + c = 0

**Examples:** 3x^{2} + x + 5 = 0, -x^{2} + 7x + 5 = 0, x^{2} + x = 0.

### Quadratic Equation Formula

The solution or roots of a quadratic equation are given by the quadratic formula:

(α, β) = [-b ± √(b^{2} – 4ac)]/2ac

**1.** The roots of the quadratic equation: x = (-b ± √D)/2a, where D = b^{2} – 4ac

**2.** Nature of roots:

- D > 0, roots are real and distinct (unequal)
- D = 0, roots are real and equal (coincident)
- D < 0, roots are imaginary and unequal

**3.** The roots (α + iβ), (α – iβ) are the conjugate pair of each other.

**4.** Sum and Product of roots: If α and β are the roots of a quadratic equation, then

- S = α+β= -b/a = coefficient of x/coefficient of x
^{2} - P = αβ = c/a = constant term/coefficient of x
^{2}

**5.** Quadratic equation in the form of roots: x^{2} – (α+β)x + (αβ) = 0

**6.** The quadratic equations a_{1}x_{2} + b_{1}x + c_{1} = 0 and a_{2}x_{2} + b_{2}x + c_{2} = 0 have;

- One common root if (b
_{1}c_{2}– b_{2}c_{1})/(c_{1}a_{2}– c_{2}a_{1}) = (c_{1}a_{2}– c_{2}a_{1})/(a_{1}b_{2}– a_{2}b_{1}) - Both roots common if a
_{1}/a_{2}= b_{1}/b_{2}= c_{1}/c_{2}

**7.** In quadratic equation ax^{2 }+ bx + c = 0 or [(x + b/2a)^{2} – D/4a^{2}]

- If a > 0, minimum value = 4ac – b
^{2}/4a at x = -b/2a. - If a < 0, maximum value 4ac – b
^{2}/4a at x= -b/2a.

**8.** If α, β, γ are roots of cubic equation ax^{3} + bx^{2} + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a

**9.** A quadratic equation becomes an identity (a, b, c = 0) if the equation is satisfied by more than two numbers i.e. having more than two roots or solutions either real or complex.

## Roots of Quadratic Equation

The values of variables satisfying the given quadratic equation are called its roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0.

The real roots of an equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersect the x-axis.

- One of the roots of the quadratic equation is zero and the other is -b/a if c = 0
- Both the roots are zero if b = c = 0
- The roots are reciprocal to each other if a = c

### What is Discriminant?

The term (b^{2} – 4ac) in the quadratic formula is known as the discriminant of a quadratic equation. The discriminant of a quadratic equation reveals the nature of roots.

### Nature of Roots of Quadratic Equation

If the value of discriminant = 0 i.e. b^{2} – 4ac = 0 | The quadratic equation will have equal roots i.e. α = β = -b/2a |

If the value of discriminant < 0 i.e. b^{2} – 4ac < 0 | The quadratic equation will have imaginary roots i.e α = (p + iq) and β = (p – iq). Where ‘iq’ is the imaginary part of a complex number |

If the value of discriminant (D) > 0 i.e. b^{2} – 4ac > 0 | The quadratic equation will have real roots |

If the value of discriminant > 0 and D is a perfect square | The quadratic equation will have rational roots |

If the value of discriminant (D) > 0 and D is not a perfect square | The quadratic equation will have irrational roots i.e. α = (p + √q) and β=(p – √q) |

If the value of discriminant > 0, D is a perfect square, a = 1 and b and c are integers | The quadratic equation will have integral roots |

### How to Determine the Nature of the Roots of a Quadratic Equation?

**Example 1: Find the values of k for which the quadratic expression (x – a) (x – 10) + 1 = 0 has integral roots.**

**Solution:**

The given equation can be rewritten as, x^{2} – (10 + k)x + 1 + 10k = 0.

D = b^{2} – 4ac = 100 + k^{2} + 20k – 40k = k^{2} – 20k + 96 = (k – 10)^{2} – 4

The quadratic equation will have integral roots, if the value of discriminant > 0, D is a perfect square, a = 1 and b and c are integers.

i.e. (k – 10)^{2} – D = 4

Since discriminant is a perfect square. Hence, the difference of two perfect square in R.H.S will be 4 only when D = 0 and (k – 10)^{2} = 4.

⇒ k – 10 = ± 2. Therefore, the values of k = 8 and 12.

**Example 2: Find the values of k such that the equation p/(x + r) + q/(x – r) = k/2x has two equal roots.**

**Solution:**

The given quadratic equation can be rewritten as:[2p + 2q – k]x^{2} – 2r[p – q]x + r^{2}k = 0

For equal roots, the discriminant (D) = 0, i.e. b^{2} – 4ac = 0

Here, a = [ 2p + 2q – k ], b = – 2r [ p – q ] and c = r^{2}k[-2r (p – q)]^{2} – 4[(2p + 2q – k) (r^{2}k)] = 0

r^{2}(p – q)^{2} – r^{2}k(2p + 2q – k) = 0

Since r ≠ 0, therefore, (p – q)^{2} – k(2p + 2q – k) = 0

k^{2} – 2(p + q)k + (p – q)^{2}

k = 2(p+q) ± √[4(p + q)^{2} – 4(p – q)]^{2}/2 = -(p + q) ± √4pq

∴ The values of k = (p + q) ± 2√pq = (√p ± √q)^{2}

**Example 3: Find the quadratic equation with rational coefficients when one root is 1/(2 + √5).**

**Solution:**

If the coefficients are rational, then the irrational roots occur in conjugate pairs. Therefore, if one root is α = 1/(2 + √5) = √5 – 2, then the other root will be β = 1/(2 – √5) = -√5 – 2.

Sum of the roots α + β = -4 and product of roots α β = -1.

Thus, the required equation is x^{2} + 4x – 1 = 0.

**Example 4: Form a quadratic equation with real coefficients when one of its root is (3 – 2i).**

**Solution:**

Since the complex roots always occur in pairs, so the other root is 3 + 2i. Therefore, by obtaining the sum and the product of the roots, we can form the required quadratic equation.

The sum of the roots is

(3 + 2i) + (3 – 2i) = 6. The product of the root is (3 + 2i) × (3 – 2i) = 9 – 4i^{2} = 9 + 4 = 13.

Hence, the equation is x^{2} – Sx + P = 0

Therefore, x^{2} – 6x + 13 = 0 is the required quadratic equation.

## Relationships between Coefficient and Roots of Quadratic Equation

If α and β are roots of a Quadratic Equation ax^{2} + bx + c then,

- α + β = -b/a
- αβ = c/a
- α – β = ±√[(α + β)
^{2}– 4αβ] - |α + β| = √D/|a|

The relationship between the roots and coefficient of a polynomial equation can be derived by simplifying the given polynomials and substituting the above results as shown below.

- α
^{2}β + β^{2}α = αβ (α + β) = – bc/a^{2} - α
^{2}+ αβ + β^{2}= (α + β)^{2}– αβ = (b^{2}– ac)/a^{2} - α
^{2}+ β^{2}= (α – β)^{2}– 2αβ - α
^{2}– β^{2}= (α + β) (α – β) - α
^{3}+ β^{3}= (α + β)^{3}+ 3αβ(α + β) - α
^{3}– β^{3}= (α – β)^{3}+ 3αβ(α – β) - (α/β)
^{2}+ (β/α)^{2}= α^{4}+ β^{4}/α^{2}β^{2}

**Example: If the coefficient of x in the quadratic equation x ^{2} + bx + c =0 was taken as 17 in place of 13, its roots were found to be -2 and -15. Find the roots of the original quadratic equation.**

**Solution:**

Since there is no change in the coefficient of x^{2} and c, therefore, the product of zeros will remain the same for both equations.

Therefore, the product of zeros (c) = -2 × -15 = 30,

Since, the original value of b is 13.

∴ Sum of zeros = -b/a = -13.

Hence, the original quadratic equation is:

x^{2} – (Sum of Zeros)x + (Product of Zeros) = 0

x^{2} + 13x + 30 = 0

∴ (x + 10) (x + 3) = 0

Therefore, the roots of the original quadratic equations are -3 and -10.

### Condition for Common Root(s)

## Quadratic Equations having Common Roots

Let β be the common root (solution) of quadratic equations a_{1}x^{2} + b_{1}x+c_{1} and a^{2}x_{2}+b_{2}x+c_{2}. This implies that a_{1}β^{2 }+ b_{1}β + c_{1 }= 0 and a_{2}β^{2 }+ b_{2}β + c_{2 }= 0.

Now, Solving for β^{2} and β we will get:

β^{2}/(b_{1}c_{2} – b_{2}c_{1}) = -β/(a_{1}c_{2} – a_{2}c_{1}) = 1/(a_{1}b_{2} – a_{2}b_{1}) [using determinant method]

Therefore, β^{2 }= (b_{1}c_{2} – b_{2}c_{1})/ (a_{1}b_{2} – a_{2}b_{1}) . . . . . . . . . . . . . . . . (1)

And, β = (a_{2}c_{1} – a_{1}c_{2})/(a_{1}b_{2} – a_{2}b_{1}) . . . . . . . . . . . . . . . . (2)

On squaring equation (2) and equating it with equation (1) we get:

(a_{1}b_{2} – a_{2}b_{1})/(b_{1}c_{2} – b_{2}c_{1}) = (a_{2}c_{1} – a_{1}c_{2})^{2}

Hence, it is the required condition for quadratic equations having one common root.

If both the roots of quadratic equations a_{1}x_{2} + b_{1}x + c_{1} and a_{2}x_{2} + b_{2}x + c_{2} are common then:

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}

If α is a repeated root, i.e., the two roots are α, α of the equation f(x) = 0, then α will be a root of the derived equation

f’(x) = 0 where f’(x) = df/dx

If α is a repeated root common in f(x) = 0 and ϕ(x) = 0, then α is a common root both in f’(x) = 0 and ϕ ‘(x) = 0.

### How to Solve Quadratic Equations with Common Roots?

**Example: For what value of k, both the quadratic equations 6x ^{2} – 17x + 12 = 0 and 3x^{2} – 2x + k = 0 will have a common root.**

**Solution:**

If one of the root of quadratic Equations a_{1}x^{2} + b_{1}x + c_{1} and a_{2}x^{2} + b_{2}x + c_{2} is common then: (a_{1}b_{2} – a_{2}b_{1})(b_{1}c_{2} – b_{2}c_{1}) = (a_{2}c_{1} – a_{1}c_{2})^{2} . . . . . . . . . . . . . (1)

Form the given quadratic Equations, a_{1} = 6, b_{1} = -17, c_{1} = 12, a_{2} = 3, b_{2} = -2 and c_{2} = k

On substituting these values in equation (1), we will get:[(6×-2) – (3×-17)] × [-17k – (-2×12)] = (3×12 – 6k)^{2}

-663k + 936 = 1296 + 36k^{2} – 432k

36k^{2} + 231k + 360 = 0

12k^{2} + 125k + 120 = 0

(4k + 15) (3k + 8) = 0

Therefore, the values of k are -154, -83.

**Example: Find the values of k such that the Quadratic Equations x ^{2} – 11x + k and x^{2} – 14x + 2k have a common factor.**

**Solution:**

Let (x – α) be the common factor of quadratic Equations x^{2} – 11x + k and x^{2} – 14x + 2k Then x = α will satisfy the given quadratic equations.

Therefore, α^{2} – 11α + k = 0 . . . . . . . . . . (1)

And, α^{2} – 14α + 2k = 0 . . . . . . . . . . . . (2)

On Solving Equation (1) and Equation (2) we will get:

α^{2}/(-22k + 14k) = -α/2k – k = 1/(-14 + 11)

Therefore, α^{2 }= (-22k + 14k)/-3 = 8/3 . . . . . . . (3)

And, α = (2k – k)/(-14 + 11) = k/3 . . . . . . . . . . . (4)

On Equating Equation (3) and Equation (4):

8/3 = (k/3)^{2}

Therefore, the value of k = 24.

## Solving Quadratic Equations

There are two methods to solve a quadratic equation

**Algebraic Method****Graphical Method**

### Algebraic Method of Solving Quadratic Equations

**General Form:** ax^{2} + bx + c = 0;

x^{2} + bx/a + c/a = 0

⇒ (x + b/2a)^{2} = b^{2}/4a^{2} – c/a

Or, (x + b/2a)^{2} = (b^{2} – 4ac)/4a^{2}

Or, x + b/2a = ± (√b^{2} – 4ac)/2a

⇒ x = [-b ± √(b^{2} – 4ac)]/2a

b^{2} – 4ac = Discriminant (D)

- α = (-b+√D)/2a
- β = (-b – √D)/2a

α+β= -b/a, α.β = c/a

Therefore, the quadratic equation can be written as,

⇒ x^{2} – (α + β)x + (α.β) = 0.

## Tips to Solve Equations reducible to Quadratic

- To solve the equations of type ax
^{4}+ bx^{2}+ c = 0, put x^{2}= y - To solve a.p(x)
^{2 }+ b.p(x) + c = 0, put p(x) = y. - To solve a.p(x) + b/p(x) + c = 0, put p(x) = y.
- To solve a(x
^{2}+ 1/x^{2}) + b(x + 1/x) + c = 0,put x + 1/x = y and to solve a(x^{2}+ 1/x^{2}) + b(x – 1/x) + c = 0, put x – 1/x = y. - To solve a reciprocal equation of the type ax
^{4}+ bx^{3}+ cx^{2}+ bx + a = 0, a ≠ 0, divide the equation by d^{2}y/dx^{2}to obtain a(x^{2 }+ 1/x^{2}) + b(x + 1/x) + c = 0,and then put x + 1/x = y. - To solve (x + a) (x + b) (x + c) (x + d) + k = 0 where a + b = c + d, put x
^{2}+(a + b)x = y - To solve an equation of type √(ax + b) = cx + d or √(ax
^{2}+ bx + c) = dx + e, square both the sides. - To solve √(ax + b) ± √(cx + d) = e, transfer one of the radical to the other side and square both the sides. Keep the expression with radical sign on one side and transfer the remaining expression on the other side.

## Graphical Solution of Quadratic Equation

Consider a quadratic equation ax^{2} + bx + c = 0, where a, b, and c are real and a ≠ 0. The expression can be further rewritten as:

a[(x + b/2a)^{2} + (D/4a^{2})]

The above quadratic equation represents a parabola whose vertex is at P [-b/2a, -D/4a] and axis parallel to y-axis.

In a quadratic equation, the value of ‘a’ determines whether the graph of a quadratic equation will be concave upwards (a > 0) or concave downwards (a < 0). The value of discriminant (b^{2} – 4ac) determines whether the graph of a quadratic equation will:

- Intersect the x-axis at two points i.e. b
^{2}– 4ac > 0 - Just touches the x-axis i.e. b
^{2}– 4ac = 0 - Never intersects the x-axis i.e. b
^{2}– 4ac < 0

### Graphing Quadratic Equations

**The quadratic equation representing a parabola with vertex at P and axis parallel to the y-axis**

**Case 1: When a > 0 and b ^{2} – 4ac > 0**

The graph of a quadratic Equation will be concave upwards and will intersect the x-axis at two points α and β with α < β. The quadratic equation will have two real roots (α and β) and the curve will always lie above the x-axis.

- The quadratic function f(x) will be positive i.e. f(x) > 0, for the values of x lying in the interval (-∞, α) ∪ (β, ∞)
- The quadratic function f(x) will be Equal to zero i.e. f(x) = 0, if x = α or β
- The quadratic function f(x) will be negative i.e. f(x) < 0, for the values of x lying in the interval (α, β)

**Case 2: When a > 0 and b ^{2} – 4ac = 0**

The graph of a quadratic equation will be concave upwards and will touch x-axis at a point -b/2a. The quadratic equation will have two equal real roots i.e. α = β. The quadratic function f(x) will be positive i.e. 0 ≤ f(x), x ∈ R.

**Case 3: When a > 0 and b ^{2} – 4ac < 0**

The graph of a quadratic equation will be concave upwards and will not intersect the x-axis. The quadratic equation will have imaginary roots and the curve will always lie above the x-axis. The quadratic function f(x) will be positive i.e. f(x) > 0, x ∈ R.

**Case 4: When a < 0 and b ^{2} – 4ac > 0**

The graph of a quadratic equation will be concave downwards and will intersect the x-axis at two points α and β with α < β. The quadratic equation will have two real roots (α and β) and the curve will always lie below the x-axis.

- The quadratic function f(x) will be positive i.e. f(x) > 0, for the values of x lying in the interval (α, β).
- The quadratic function f(x) will be equal to zero i.e. f(x) = 0, if x = α or β
- The quadratic function f(x) will be negative i.e. f(x) < 0 for the values of x lying in the interval (−∞, α) ∪ (β, ∞)

**Case 5: When a < 0 and b ^{2} – 4ac = 0**

The graph of a quadratic equation will be concave downwards and will touch x-axis at a point -b/2a. The quadratic equation will have equal real roots i.e. α = β. The quadratic function f(x) will be negative i.e. f(x) ≤ 0, x ∈ R.

**Case 6: When a < 0 and b ^{2} – 4ac < 0**

The graph of a quadratic equation will be concave downwards and will not intersect the x-axis. The quadratic equation will have imaginary roots and the curve will always lie below the x-axis. The quadratic function f(x) will be negative i.e. f(x) < 0, x ∈ R.

37,513

## Range of Quadratic Equation

Consider a quadratic expression f(x) = ax^{2} + bx + c, where a ≠ 0 and a, b, and c are real. The quadratic expression can be further rewritten as f(x) = x^{2} + bax = ca.

**Intervals in which the roots of Quadratic Equation lie**

**Case 1:** Both the roots of a quadratic equation (α, β) are greater than any given number ‘m’ if,

- b
^{2}– 4ac = (D) ≥ 0, - -b/2a > m,
- f (m) > 0.

**Case 2:** Both the roots of a quadratic expression (α, β) are less than any given number ‘m’ if,

- b
^{2 }– 4ac = (D) ≥ 0, - -b/2a < m,
- f(m) > 0.

**Case 3:** Both the roots of a quadratic expression (α, β) will lie in the given interval (m1, m2) if,

- b
^{2 }– 4ac = (D) ≥ 0, - m1 < -b/2a > m2,
- f(m1) > 0,
- f(m2) > 0.

**Case 4:** If exactly one root of a quadratic equation (α, β) will lie in the given interval (m1, m2) if, f(m1).f(m2) < 0**Case 5:** The given number ‘m’ will lie between the roots of a quadratic Equation α and β if, f(m) < 0

**Case 6:** The roots of a quadratic equation α and β will have an opposite sign if, f(0) < 0.

**Case 7:** Both the roots of a quadratic expression α and β are positive if,

- b
^{2}– 4ac = (D) ≥ 0, - α + β = -b/a > 0,
- αβ = c/a > 0.

Case 8: Both the roots of a quadratic expression α and β are negative if,

- b
^{2}– 4ac = (D) ≥ 0, - 2. α + β = -b/a < 0,
- 3. αβ = c/a < 0.

### How to Find Range of Quadratic Equation?

**Example 1: Find the range of k for which 6 lies between the roots of the quadratic equation x ^{2 }+ 2(k – 3)x + 9 = 0.**

**Solution:**

6 will lie between the roots of the quadratic expression f(x) = x^{2 }+ 2(k – 3)x + 9 if,

f (6) < 0

i.e. 36 + 2 (k – 3) . 6 + 9 < 0,

= 36 + 12k – 36 + 9 < 0,

= k < -34

Therefore, the range of k for which 6 lies between the roots of the given quadratic equation is:

k ∈ (-∞, -34)

**Example 2: Find the values of k for which exactly one root of the quadratic equation 4x ^{2} – 4(k – 2)x + k – 2 = 0 lies in (0,12).**

**Solution:**

Exactly one root of the quadratic expression 4x^{2} – 4(k – 2)x + k – 2 = 0 will lie in the given interval if,

f(m1) . f(m2) < 0

i.e. f(0) . f(12)<0

Or, (k – 2) [1 – 2 (k – 2) + k – 2] < 0,

Or, (k -2) (3 – k) < 0,

Or, (k – 2) (k – 3) > 0.

Therefore, the values of k for which exactly one root of the given quadratic equation will lie in the interval (0,12):

k ∈ (-∞, 2) ∪ (3, ∞)

**Example 3. Find the values of k for which the roots of the quadratic equation x ^{2} – (k – 3)x + k = 0 are greater than 2.**

**Solution:**

The roots of the given quadratic expression f(x) = x^{2} – (k – 3)x + k are greater than 2 if,

**Condition 1:** -b/2a > 2

i.e. +(k – 3)/2 >2

Therefore, k > 7 . . . . . . . . . . . . . (1)

**Condition 2:** f (2) > 0

i.e. 4 – (k – 3)^{2} + k > 0,

Or, 4 – 2k + 6 + k > 0.

Therefore, k < 10 . . . . . . . . . . . . . . (2)

**Condition 3:** b^{2} – 4ac ≥ 0

i.e. (k – 3)^{2} – 4k ≥ 0,

Or, k^{2} + 10k + 9 ≥ 0,

Or, (k – 9) (k – 1) ≥ 0.

Therefore, k ∈ (-∞, 1] ∪ [9, ∞) . . . . . . . . . (3)

From Equation (1), (2) and, (3) the range of ‘k’ for which the roots of the given quadratic expression are greater than 2: k ∈ [9, 10).

## Maximum and Minimum Value of Quadratic Equation

**Conditions for Minimum and Maximum Value of Quadratic Equation**

To find the minimum value of a quadratic equation we need to understand the nature of the graph of these equations for different values of ‘a’. The graph of the quadratic equation f(x) = ax^{2} + bx + c will be either concave upwards (a>0) or concave downwards (a<0) respectively.

When the graph is concave upwards then its vertex determines the minimum value of a quadratic function f(x) and when it is concave downwards, its vertex determines the maximum values of a quadratic function f(x).

### How to Find Maximum and Minimum values of Quadratic Functions?

**Case 1:** When a > 0, the absolute range of a quadratic expression is given by:[-b^{2} – 4ac/4a, ∞) or [-D/4a, ∞)

**Case 2:** When a < 0, the absolute range of a quadratic equation is given by:

(-∞, -D/4a]

Also, the maximum and minimum values of a quadratic equation f(x) occurs at x = -b/2a.

If the given quadratic equation is in the form f(x) = a(x – h)^{2} + m, Then the value of ‘m’ (vertex) gives us the minimum (when ‘a’ is negative) or maximum (when ‘a’ is positive) values of the given function.

### Maximum and Minimum Values in Restricted Domain

**Case 1:** -b/2a ∉ [m, n]

f(x) = ∈ [ minimum {f(m), f(n)}, maximum {f(m), f(n)} ]

**Case 2:** -b/2a ∈ [m, n]

f(x) ∈ [minimum {f(m), f(n) – D/4a}, maximum {f(m), f(n), -D/4a}]

**Example 1: Find the maximum or minimum value of quadratic equation -4(x – 2) ^{2 }+ 2.**

**Solution:**

Since the value of ‘a’ is negative, therefore the given quadratic equation will have a maximum value. Hence, the maximum value of the quadratic equation -4(x – 2)^{2} + 2 is 2.

**Example 2: Find the minimum and maximum values of quadratic equation f(x) = x ^{2} – 12x + 11.**

**Solution:**

Since a > 0, the maximum and minimum values of a quadratic expression is given by:[(-b^{2 }– 4ac)/4a, ∞) or [-D/4a, ∞)

Therefore, the minimum value of f(x) is:

-(144 – 44)/4 at x = -(-12/2) = -25 at x = 6.

The maximum value of f(x) is infinity.

Therefore, the range of the given quadratic equation is [- 25, ∞).

**Example 3: Find the minimum value of equation (2x – 5)/(2x ^{2} + 3x + 6)?**

**Solution:**

f(x) will have minimum value at x = -b/2a [Since a>0]

i.e. at x = -3/4.

∴ the minimum value of quadratic equation f(x) = -D/4a = -(9 – 48)/8 = 39/8.

**Example 4: Find the range of function f(x) = (x+2)/(2x ^{2} + 3x + 6), if x is real.**

**Solution:**

Let y = 2x^{2 }+ 3x + 6

2x^{2}y + 3xy + 6y = x + 2

2x^{2}y + (3y – 1)x + 6y -2 = x + 2

Now, discriminant = (3y – 1)^{2} – 4(2y).(6y – 2) ≥ 0 [because x is real]

9y^{2} + 1 – 6y – 48y^{2} + 16y ≥ 0

39y^{2} – 10y – 1 ≤ 0

39y^{2} – 13y + 3y – 1 ≤ 0

13y(3y – 1) + 1(3y – 1) ≤ 0

∴ The range of quadratic function f(x) = (x+2)/(2x^{2} + 3x + 6): y ∈ [-1/13, 1/3].

**⇒ Try this:**

If both the roots of the quadratic equation x^{2 }+ x(4 – 2k) + k^{2} – 3k – 1 = 0 are less than 3, then find the range of values of k.

**Ans:** k ∈ (-∞, 4).

## Descartes Rule of Signs for Quadratic Polynomials

This is a very famous rule that helps in getting an idea about the roots of a polynomial equation. The rule states that the number of positive real roots of P_{n}(x) = 0 cannot be more than the number of sign changes. Similarly, the number of negative roots cannot be more than the number of sign changes in P_{n}(-x).

**Example 1: How many real roots does the equation 2x ^{5} + 2x^{4} – 11x^{3 }+ 9x^{2} – 4x + 2 = 0 will have?**

**Solution:**

The given equation has 4 sign changes so it can have a maximum of 4 positive real roots. Now for f(-x), the equation has only one sign change i.e. f (-x) = -x^{5} + 2x^{4} + x^{3} + x^{2} + x + 2 = 0. Hence, the equation will have only one negative real root.

**Example 2: How many real roots do the quadratic equation x ^{2 }+ 3|x| + 2 will have?**

**Solution:**

Since the quadratic equation has no sign change for both f(x) and f(-x). Therefore, the equation will have no real root.

### Sign Convention of Quadratic Equation – ax^{2} + bx + c = 0

- The roots of quadratic equation are equal in magnitude but of opposite sign if b = 0 and ac < 0
- The root with greater magnitude is negative if the sign of a = sign of b × sign of c
- If a > 0, c < 0 or a > 0, c > 0; the roots of quadratic equation will have opposite sign
- If y = ax
^{2}+ bx + c is positive for all real values of x, a > 0 and D < 0 - If y = ax
^{2}+ bx + c is negative for all real values of x, a < 0 and D < 0

**Example: For what values of ‘m’, the quadratic expression x ^{2} + 2(m + 1)x + 9m – 5 = 0 will have only negative roots.**

**Solution:**

Here a = 1, b = 2(m + 1), and c = 9m – 5

Therefore, D = 4(m + 1)2 – 4(9m – 5) = 4m^{2} + 24 – 28m . . . . . (1)

α + β = -2 (m + 1) and αβ = 9m – 5 . . . . . . . . . . . . . . (2)

Now, both the roots of a quadratic expression are negative if, b^{2} – 4ac = (D) ≥ 0, α + β = −ba < 0, and αβ = ca < 0

Therefore, 4m^{2} + 24 – 28m ≥ 0 [From equation (1)]

i.e. m^{2} – 7m + 6 ≥ 0

Or, (m – 1) (m – 6) ≥ 0

Therefore, m ≤ 1 or m ≥ 6 . . . . . . . . . . (3)

Now, α + β < 0

i.e. -2(m + 1) < 0 [From equation (2)]

Or, m > -1 . . . . . . . (4)

And, α β < 0

i.e. 9m – 5 < 0

Or, m < 59 . . . . . . (5)

From equations (3), (4), and (5) we can conclude that the given quadratic equation will have only negative roots if m ≥ 6.

## Quadratic Equations in Two Variables

The general form of a quadratic equation ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c can be factorized into two linear factors as shown below,

ax^{2} + 2(hx + g)x + by^{2} + 2fy + c = 0 . . . . (i)

Using the quadratic formula in above equation we get,

x = [2(hy + g) ± √{4(hy + g)^{2} – 4a(by^{2} + 2fy + c)}]/2a

ax + hy + g = ± √(h^{2}y^{2} + g^{2} + 2ghy – aby^{2} – 2afy – ac) . . . . (ii)

At this point, the expression (i) can be resolved into two linear factors if,

(h^{2} – ab)y^{2} + 2(gh – af)y + g^{2} – ac is a perfect square and h^{2} – ab > 0

But (h^{2} – ab)y^{2} + 2(gh – af)y + g^{2} – ac will be a perfect square if discriminant = 0

i.e. g^{2}h^{2} + a^{2}f^{2} – 2afgh – h^{2}g^{2} + abg^{2} + ach^{2} – a^{2}bc = 0 and h^{2} – ab > 0

abc + 2fgh – af^{2} – bg^{2} – ch^{2} = 0 and h^{2} – ab > 0

This is the required condition. The condition that this expression may be resolved into two linear rational factors is:

abc + 2fgh – af^{2} – bg^{2} – ch^{2} = 0 and h^{2} – ab > 0

This expression is called the discriminant of the above quadratic expression.

## Biquadratic Equation

A polynomial equation of degree four without the terms of degree three and one is known as a biquadratic equation.

**General Form:** z^{4} + a_{0}z^{2} + c = 0

**Examples:**

- x
^{4}+16x^{2}+ 6 = 0 - 2x
^{4}+ 14x^{2}+ 3 = 0

### How to Solve Biquadratic Equations?

Biquadratic polynomials can be easily solved by converting them into quadratic equations i.e. by replacing the variable ‘z’ by x^{2}.

**Example: Find the zeros of a biquadratic equation x ^{4 }– 3x^{2} + 2 = 0.**

**Solution:**

Given f (x) = x^{4 }– 3x^{2} + 2

On substituting x^{2} = z in the given equation we get,

f(x) = z^{2} – 3z + 2 = 0

z^{2} – 2z – z + 2 = 0

z(z – 2) -1(z – 2) = 0

∴ z = 1 and z = 2

Hence, x = ±√1 and x = ±√2 [Since, z = x²].

## Practice Questions for Solving Quadratic Equations

**Example 1: If G(x) = px ^{2} + qx + r and H(x) = -px^{2} + mx + r and p.r ≠ 0. Prove that G(x) . H(x) = 0 will have exactly two real roots.**

**Solution:**

G(x) will have real roots if q^{2} – 4pr ≥ 0 and H(x) will have real roots if m^{2} + 4pr ≥ 0.

**Case 1:** If pr < 0,

q^{2} – 4pr ≥ 0, therefore, G(x) will have real root(s).

**Case 2:** If pr > 0,

m^{2} + 4pr ≥ 0, therefore, H(x) will also have real root(s).

Hence at least two roots of G(x).H(x) are real.

**Example 2: Find the range in which the value of given quadratic expression will lie (x ^{2} + 14x + 9)/(x^{2} + 2x + 3), where x is a real number.**

**Solution:**

Let, y = (x^{2} + 14x + 9)/(x^{2} + 2x + 3)

i.e. x^{2} + 14x + 9 = (x^{2}y + 2xy + 3y)

Or, (1 – y)x^{2} + (7 – y)2x + 3(3 – y) = 0

Since x is real, therefore, b^{2} – 4ac ≥ 0

i.e. [2(7 – y)]^{2} – 4[1 – y] × 3[3 – y]≥0

Or, -y^{2} – y + 40 ≤ 0

Or, (y + 5) (y – 4) ≤ 0

Therefore, the range in which the value of the given quadratic expression will lie is -5 ≤ y ≤ 4.

**Example 3: Find the values of x for which the expression (x ^{2} – 4x + 3)/(x^{2} + x + 1) ≤ 0.**

**Solution:**

Let f(x) = x^{2} – 4x + 3 and g(x) = x^{2} + x + 1.

The coefficient of x^{2} in g(x) is positive and the value of discriminant (D) < 0. Hence, g(x) is positive for all values of x.

Since, f(x)/g(x) < 0. Therefore, f (x) must be less than 0.

i.e. x^{2} – 4x + 3 < 0

Or, (x – 3) (x – 1) < 0

Therefore, 1 < x < 3.

**Example 4: Find the values of ‘m’ for which the roots of the given quadratic equation f(x) = 9x ^{2 }+ (m – 4)x + m/4 satisfy the following given conditions:**

**Both the roots of quadratic polynomial f(x) are real and distinct.****Quadratic expression f(x) has equal roots.****Roots of quadratic expression are not real.****Roots of the quadratic equation have opposite sign.****Roots are equal in magnitude but opposite sign.****Both the roots of the given quadratic equation f(x) are positive.****Both of the roots are negative.**

**Solution:**

From the given quadratic equation f(x) = 9x^{2 }+ (m – 4)/x + m/4,

a = 9, b = (m – 4) and c = m/4

Therefore, D = b^{2 }– 4ac = (m – 4)^{2} – 4(9).(m/4)

= m^{2} + 16 – 8m – 9m = m^{2 }– 17m + 16

Therefore, D = m^{2} – 17m + 16 = (m – 1) (m – 16)

**Solution 1.** Both the roots of the quadratic equations are real and distinct if,

D = (b^{2 }– 4ac) > 0

i.e (m – 16) (m – 1) > 0

**Note:** If ax^{2 }+ bx + c > 0, D > 0 and a > 0, then x ∈ (-∞, α) ∪ (β, ∞).

Therefore, both the roots of given quadratic equation are real and distinct if:

m ∈ (-∞, 1) ∪ (16, ∞).

**Solution 2.** The quadratic expression f(x) has equal roots if, D = 0

i.e (m – 16) (m – 1) = 0

Therefore, for m = 16 or m = 1, both the roots of given quadratic equation are equal.

**Solution 3.** Roots of quadratic equation are imaginary if, D < 0

i.e. (m – 16) (m – 1) < 0

**Note:** If ax^{2 }+ bx + c > 0, D < 0 and a > 0, then x ∈ R

Therefore, the roots of the given equation are imaginary if, m ∈ (1, 16).

**Solution 4.** Roots of quadratic expression will have opposite sign if, f (0) < 0

Since f(0) = m

Therefore, the roots of the given equation will have opposite sign if m ∈ (−∞,0).

**Solution 5.** The roots of quadratic expression are equal in magnitude but opposite sign if,

**Condition 1:** Sum of roots = 0

i.e. (m – 4)/9 = 0, Therefore, m = 4 . . . . . . . . . . . . . . . (1)

**Condition 2:** D = (b^{2} – 4ac) ≥ 0

i.e. (m -1) (m – 16) ≥ 0

Therefore, x ∈ (=∞, 1] ∪ [16, ∞) . . . . . . . . . . . . . . . . (2)

From equation (1) and (2), we conclude that no such value of ‘m’ exist in f(x) when the roots have opposite sign with equal magnitude.

**Solution 6.** Both the roots of given quadratic expression are positive if,

**Condition 1:** Product of the roots > 0

i.e. m/(4×9) > 0, Therefore, m > 0 . . . . . . . . . . (3)

**Condition 2:** Sum of the roots > 0

-(m – 4)/9 > 0, Therefore, m < 4 . . . . . . . . . (4)

**Condition 3:** D = (b^{2} – 4ac) ≥ 0

From equation (2),

x ∈ (-∞, 1] ∪ [16, ∞) . . . . . . . . (5)

Therefore, from equation (3), (4), and (5) both the roots of f(x) are positive if, m ∈ (0,1].

**Solution 7.** Both the roots are negative if,

**Condition 1:** The Product of roots > 0

i.e. m/(4×9) > 0, m > 0 . . . . . . . . . . (6)

**Condition 2:** The Sum of roots < 0

-(m – 4)/9 < 0, m > 4 . . . . . . . . . (7)

**Condition 3:** D = (b^{2 }– 4ac) ≥ 0

Using equation (2),

x ∈ (-∞, 1] ∪ [16, ∞) . . . . . . . . (8)

Therefore, from equations (6), (7), and (8) both the roots of quadratic expression f (x) are negative if,

m ∈ [16, ∞].

**Example 5: The quadratic equations x ^{2} – ax + b = 0 and x^{2} – px + q = 0 have a common root and the second equation has equal roots, show that b + q = ap/2.**

**Solution:**

By considering α and β to be the roots of equation (i) and α to be the common root, we can solve the problem by using the sum and product of roots formula.

The given quadratic equations are

x^{2} – ax + b = 0 ………. (i)

x^{2} – px + q = 0 ………..(ii)

From equation (i), α + β = a, α = b

From equation (ii), 2α = p, α^{2} = q

b + q = αβ + α^{2} = α (α + β) = ap/2

**Example 6: x ^{2} + ax + bc = 0 and x^{2} + bx + ca = 0 have a non zero common root and a ≠ b. Show that the other roots are roots of the quadratic equation x^{2} + cx + ab = 0, c ≠ 0.**

**Solution:**

By considering α to be the common root of the quadratic equations and β, γ to be the other roots of the equations respectively, then by using the sum and product of roots formula we can prove this.

Further, α + β = -a and αβ = bc;

α + γ = -b, αγ = ca

2α + β + γ = -(a + b) and α^{2}βγ = abc^{2} . . . . . . (i)

Therefore, β + γ = c – 2c = -c . . . . . . . (ii)

and, c^{2}βγ = c^{2}ba

Therefore, βγ = ab …. (iii)

From equation (ii) and (iii),

β and γ are the roots of the quadratic equation x^{2} + cx + ab = 0.

**Example 7: The expressions x ^{2} – 11x + a = 0 and x^{2} – 14x + 2a = 0 must have a common factor and a ≠ 0, find the common factor and the common root.**

**Solution:**

If (x – α) is the common factor of the given quadratic equations then, x = α becomes the root of the corresponding equation.

Therefore, α^{2} – 11α + a = 0 and α^{2} – 14α + 2a = 0

On subtracting the above equations we get,

3α – a = 0 ⇒ α = a/3

Hence, a^{2}/9 – 11a/3 + a = 0,

On solving the above quadratic equation we get, a = 0 or 24

Since a ≠ 0, therefore, a = 24.

**Example 8: Determine the values of m for which the quadratic equations 3x ^{2} + 4mx + 2 = 0 and 2x^{2} + 3x – 2 = 0 may have a common root.**

**Solution:**

Consider α to be the common root of the given equations.

Then, 3α^{2} + 4mα + 2 = 0 and 2α^{2} + 3α – 2 = 0

Using the cross multiplication method, we have

(-6 – 4)^{2} = (9 – 8m) (-8m – 6)

⇒ 50 = (8m – 9)(4m + 3)

Or, 32m^{2} – 12m – 77 = 0

Or, (8m + 11) (4m – 7) = 0

Therefore, the values of m for which the given quadratic equations will have common roots = -11/8 and 7/4.

**Example 9: Find the relation between c, a and m if a quadratic equation formed from y ^{2} = 4ax and y = mx + c has equal roots.**

**Solution:**

(mx + c)^{2} = 4axm^{2}x^{2} + 2(cm – 2a)x + c^{2} = 0

Since, the roots of this quadratic equation are equal. Therefore, D = 04(cm – 2a)^{2} – 4m^{2}c^{2} = 0

4a^{2} – 4cma = 0 or a^{2} = acm

⇒ a = cm or c = a/m.

This is a condition for the line y = mx + c to be tangent to the curve y^{2} = 4ax.

**5. For what values of ‘m’ does the quadratic equation (1 + 2) x ^{2} – 2(1 + 3m)x + (1 + 8m) = 0 have equal roots?**

**Solution:**

The roots are equal if discriminant (D) = 4(1 + 3m)^{2} – 4(1 + m)(1 + 8m) = 0

m^{2} – 3m = 0 ⇒ m = 0, 3.