a^2 + 5a - 24 = 0 complete square method.

Question

Solve the following quadratic equation by completing square method

a² + 5a - 24 = 0.

Answer 1

We have, a² + 5a - 24 = 0

∴ a² + 5a = 24   .....(i)

Third term = (\(\frac 12\)​ coefficient of x)²

\(= (\frac 12 \times 5)^2\)

\(= \frac{25}4\)

Adding \( \frac{25}4\) to both sides of equation (i), we get

\(a^2 + 5a + \frac{25}4 = 24 + \frac{25}4\)

4a² + 20a + 25 = 96 + \(\frac{25}4\)

(2a)² + 20a + (5)² = \(\frac{121}4\)

(2a + 5)² = (\(\frac{11}2\))²

Taking square roots on both sides

2a + 5 = ± \(\frac{11}2\)

2a = -5 ± \(\frac{11}2\)

\(2a = -5 + \frac{11}2\)

\( 2a = \frac{-10 + 11}2\)

\(a = \frac 1{2 \times 2}\)

\(a=\frac 14\)

or

\(2a = -5 - \frac{11}2\)

\( 2a = \frac{-10 - 11}2\)

\(a = \frac {-21}{2 \times 2}\)

\(a = \frac{-21}4\)

Hence \(\frac 14\) and \( \frac{-21}4\) are the roots of the given quadratic equation.