NATURE OF ROOTS AND EXERCISE 2 3 - PUNJAB-CURRICULUM-AND-TEXTBOOK-BOARD-PCTB / 10TH-CLASS / MATHEMATICS / MATHEMATICS-10-10TH-CLASS-NBF-2024 / QUADRATIC-EQUATIONS / NATURE-OF-ROOTS-AND-EXERCISE-2-3

Nature of Roots of a Quadratic Equation

We have already derived the quadratic formula by solving the general form of quadratic equation for $x$ . We know that the solutions of a quadratic equation of the form $ax^{2} + bx + c = 0$ with $a \neq 0$ , are given by this formula:

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

To define discriminant, first we solve the following examples by using the quadratic formula:

Example A: Solve. $x^{2} - 3x - 28 = 0$

Solution: We have $a = 1$ , $b = - 3$ and $c = - 28$

x = \frac{- (- 3) \pm \sqrt{(- 3)^{2} - 4 (1) (- 28)}}{2 (1)}

= \frac{3 \pm \sqrt{121}}{2}

x = \frac{3 \pm 11}{2} = 7, - 4

The roots of equation are rational and unequal.

Example B: Solve $x^{2} - 8x + 16 = 0$

Solution: We have $a = 1$ , $b = - 8$ and $c = 16$

x = \frac{- (- 8) \pm \sqrt{(- 8)^{2} - 4 (1) (16)}}{2 (1)}

= \frac{8 \pm 0}{2}

x = \frac{8 + 0}{2}, x = \frac{8 - 0}{2}

x = 4, x = 4

The roots of equation are rational and equal.

1. The above chart shows that if the value of the discriminant is a perfect square or 0, the roots are real and rational. Other positive discriminant will yield irrational roots. A negative discriminant means roots will be imaginary/complex.

Example: Find the value of discriminant and describe the nature of roots.

$2 x^{2} + x - 3 = 0 b . x^{2} + 8 = 0$

Solution:

The value of discriminant is positive and a perfect square. So, the given equation has two real roots and they are rational and unequal.

Nature of Roots and Exercise 2.3

Nature of Roots of a Quadratic Equation