Question

Solve the following quadratic equation by completing the square method.

5x² = 4x + 7

Answer 1

 5x² = 4x + 7 

\[\Rightarrow 5 x^2 - 4x - 7 = 0\]
\[ \Rightarrow x^2 - \frac{4}{5}x - \frac{7}{5} = 0\]
\[ \Rightarrow x^2 - \frac{4}{5}x + \left( \frac{\frac{4}{5}}{2} \right)^2 - \left( \frac{\frac{4}{5}}{2} \right)^2 - \frac{7}{5} = 0\]
\[ \Rightarrow \left[ x^2 - \frac{4}{5}x + \left( \frac{2}{5} \right)^2 \right] - \left( \frac{2}{5} \right)^2 - \frac{7}{5} = 0\]
\[ \Rightarrow \left( x - \frac{2}{5} \right)^2 = \frac{7}{5} + \frac{4}{25} = \frac{35 + 4}{25} = \frac{39}{25}\]
\[ \Rightarrow \left( x - \frac{2}{5} \right)^2 = \frac{39}{25}\]
\[\Rightarrow \left( x - \frac{2}{5} \right)^2 = \left( \frac{\sqrt{39}}{5} \right)^2 \]
\[ \Rightarrow x - \frac{2}{5} = \frac{\sqrt{39}}{5} \text{ or } x - \frac{2}{5} = - \frac{\sqrt{39}}{5}\]
\[ \Rightarrow x = \frac{2 + \sqrt{39}}{5} \text{ or } x = \frac{2 - \sqrt{39}}{5}\]