# CLASS-10NATURE OF A ROOTS OF A QUADRATIC EQUATIONS

Nature of Roots of a Quadratic Equations

Consider the quadratic equations ax²+ bx + c = 0, where a, b, c are real numbers and a ≠ 0. Let α and β be the roots of this equation.

Let, D = (b²- 4ac). Then, D is called the discriminant of the given equation.

A. If D > 0, the roots area real and distinct, given by

- b + √D                 - b - √D

α = ------------ and β = -------------

2a                        2a

Case 1. If D > 0 and not a perfect square, then the roots are unequal and irrational.

Case 2. If D > 0 and a perfect square, then the roots are –

(i) Rational and unequal, if a, b, c are rational

(ii) Irrational and unequal, if b is irrational

- b

B. If D = 0, the roots are real and equal, each equal to ------.

2a

C. If D < 0, the roots are unequal and imaginary.

Note : Clearly, for real roots, we must have, D ≥ 0.

Example.1) without solving following equations, find the value of ‘p’ for which the roots are real and equal. px²- 4x + 3 = 0

Ans.) The given equation is px²- 4x + 3 = 0

This is of the form ax²+ bx + c = 0, where a = p, b = - 4, and c = 3.

So, D = b²- 4ac = (- 4)²- 4 X p X 3

= 16 – 12p

For the given equation to have real and equal roots, we must have D = 0.

Now, D = 0

=> 16 – 12p = 0

=> 12p = 16

=> p = 16/12 = 4/3

Hence, the required value of p is 4/3.       (Ans.)

Example.2) Without solving the given quadratic equation, find the value of ‘p’ for which it has real and equal roots => x²+ (p – 3)x + p = 0

Ans.) The given equation is x²+ (p – 3)x + p = 0

This is of the form ax²+ bx + c = 0, where a = 1, b = (p – 3), c = p

So, D = (b²- 4ac)

For real & equal roots, we must have D = 0

So, D = 0 => {(p – 3)²- (4 X 1 X p)} = 0

=> (p²- 6p + 9 – 4p) = 0

=> p²- 10p + 9 = 0

=> p²- (9 + 1)p + 9 = 0

=> p²- 9p – p + 9 = 0

=> p(p – 9) – (p – 9) = 0

=> (p – 9) (p – 1) = 0

=> (p – 9) = 0 or (p – 1) = 0

=> p = 9 or p = 1

Hence, p = 9 or p = 1         (Ans.)

Example.3) Without solving the following quadratic equation, find the set of value of m for which the given equation has real and equal roots, x²+ 2(m -1)x + (m + 5) = 0

Ans.) The given equation is x²+ 2(m -1)x + (m + 5) = 0

This is of the form of ax²+ bx + c = 0, where a = 1, b = 2(m – 1), and c = (m + 5)

As we know D = (b²- 4ac)

So, D = [{2(m – 1)}² - {4 X 1 X (m + 5)}]

= {4(m²- 2m +1) – (4m + 20)}

= (4m²- 8m + 4 – 4m – 20) = 4m²- 12m – 16

For real & equal roots, we must have D = 0

Now, D = 0, so

=> 4m²- 12m – 16 = 0

=> 4m²- (16 – 4)m – 16 = 0

=> 4m²- 16m + 4m – 16 = 0

=> 4m(m – 4) + 4(m – 4) = 0

=> (m – 4) (4m + 4) = 0

=> 4(m – 4)(m + 1) = 0

=> (m – 4)(m + 1) = 0

=> (m – 4) = 0 or (m + 1) = 0

=> m = 4, or m = - 1

Hence, the required values of m is 4 or -1 and the required set is {4, -1} (Ans.)

Example.4) Show that the equation 3x²+ 7x + 8 = 0 is not true for any real value of x

Ans.) The given equation is 3x²+ 7x + 8 = 0. This is of the form ax²+ bx + c = 0, where a = 3, b = 7, c = 8

As we know, D = (b²- 4ac) = {(7)²- (4 X 3 X 8)}

= (49 – 96) = - 47 < 0

So, the given equation has no real roots.

So, 3x²+ 7x + 8 = 0 is not true for any real value of x (Ans.)

Example.5) Without solving the given equation comment upon the nature of its roots

(i) 3x²- 11x + 10 = 0,

Ans.) The given equation is 3x²- 11x + 10 = 0

This is the form of ax²+ bx + c = 0, where a = 3, b = -11, c = 10

As we know, D = (b²- 4ac) = {(- 11)² - (4 X 3 X 10)}

= (121 – 120) = 1 > 0

Thus, D > 0 and D is not a perfect square.

Hence, the roots are unequal and irrational.     (Ans.)

(ii) 15 x²- 12x + 3 = 0,

The given equation is 15 x²- 12x + 3 = 0

This is the form of ax²+ bx + c = 0, where a = 15, b = -12, c = 3

As we know, D = (b²- 4ac) = {(-12)²- (4 X 15 X 3)}

= (144 – 180) = - 36 < 0

Thus, D < 0, hence the given equation has no real roots.      (Ans.)

(iii) 9x²- 6x + 1 = 0

The given equation is 9 x²- 6x + 1 = 0

This is the form of ax²+ bx + c = 0, where a = 9, b = -6, c = 1

As we know, D = (b²- 4ac) = {(-6)² - (4 X 9 X 1)}

= (36 – 36) = 0

Hence, the roots of the given equation are real and equal.      (Ans.)