Solution of a Quadratic Equation by Quadratic formula — lesson. Mathematics CBSE, Class 10.

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How quadratic formula comes?

Consider the quadratic equation

$ax^2 + bx + c = 0$ , where

$a \ne 0$ .

Let us find the roots of this equation by the method of completing the square.

Divide the equation by

$a$ .

$x^{2} + \frac{b}{a} x + \frac{c}{a} = 0$

Move the constant to the right side.

$x^{2} + \frac{b}{a} x = - \frac{c}{a}$

Add the square of one half of coefficient of

$x$ on both sides.

$x^{2} + \frac{b}{a} x + {(\frac{b}{2 a})}^{2} = - \frac{c}{a} + {(\frac{b}{2 a})}^{2}$

${(x + \frac{b}{2 a})}^{2} = - \frac{c}{a} + \frac{b^{2}}{4 a^{2}}$

${(x + \frac{b}{2 a})}^{2} = \frac{b^{2} - 4 ac}{4 a^{2}}$

Taking square root on both sides.

$x + \frac{b}{2 a} = \pm \sqrt{\frac{b^{2} - 4 ac}{4 a^{2}}}$

$x + \frac{b}{2 a} = \pm \frac{\sqrt{b^{2} - 4 ac}}{2 a}$

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

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

Therefore, the roots of

$ax^2 + bx + c = 0$ are

$x = \frac{- b + \sqrt{b^{2} - 4 ac}}{2 a}$ and

$x = \frac{- b - \sqrt{b^{2} - 4 ac}}{2 a}$ .

The formula for finding the roots of the quadratic equation

$ax^2 + bx + c = 0$ is:

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

This formula is known as the quadratic formula.

Example:

1. Find the roots of

$2x^2 + 3x - 77 = 0$ by using quadratic formula.

Solution:

The given equation is

$2x^2 + 3x - 77 = 0$ .

Here,

$a = 2$ ,

$b = 3$ and

$c = -77$ .

Quadratic formula:

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

Substitute the given values in the formula.

$x = \frac{- 3 \pm \sqrt{3^{2} - 4 \times 2 \times (- 77)}}{2 \times 2}$

$x = \frac{- 3 \pm \sqrt{9 + 616}}{4}$

$x = \frac{- 3 \pm \sqrt{625}}{4}$

$x = \frac{- 3 \pm 25}{4}$

$x = \frac{- 3 + 25}{4}$ or

$x = \frac{- 3 - 25}{4}$

$x = \frac{22}{4}$ or

$x = \frac{- 28}{4}$

$x =$

$\frac{11}{2}$ or

$x = -7$

Therefore, the roots of the given equation are

$-7$ and

$\frac{11}{2}$ .

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2. Solution of a Quadratic Equation by Quadratic formula

Theory: