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}

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

x = \frac{22}{4}

x = \frac{- 28}{4}

\(x =\)

\frac{11}{2}

or \(x = -7\)

Therefore, the roots of the given equation are \(-7\) and

\frac{11}{2}

Did you find an error?

2. Solution of a Quadratic Equation by Quadratic formula