Question

Solve for x :

x² + 5x − (a² + a − 6) = 0

Answer 1

\[Here, a = 1, b = 5, c = - \left( a^2 + a - 6 \right)\]
\[Now, \]
\[D = \left( 5 \right)^2 - 4 \times 1 \times \left[ - \left( a^2 + a - 6 \right) \right]\]
\[ = 25 + 4 a^2 + 4a - 24\]
\[ = 4 a^2 + 4a + 1\]
\[ = \left( 2a + 1 \right)^2\]

\[\therefore x = \frac{- 5 \pm \sqrt{\left( 2a + 1 \right)^2}}{2} \left( x = \frac{- b \pm \sqrt{D}}{2a} \right)\]
\[ = \frac{- 5 + 2a + 1}{2}, \frac{- 5 - 2a - 1}{2}\]
\[ = \frac{2a - 4}{2}, \frac{- 2a - 6}{2}\]
\[ = a - 2, - a - 3\]