How do you solve using the completing the square method  x^2 + 7x – 9 = 0?

Mar 15, 2016

Answer: $x = - \frac{7}{2} \pm \frac{\sqrt{85}}{2}$ or $x \approx - 8.11 \mathmr{and} 1.11$

Explanation:

To solve by completing the square, first add 9 to both sides

${x}^{2} + 7 x = 9$

Then you have to manipulate the number in front of $x$, in this case 7. You divide it by two and square the result, giving you

${\left(\frac{7}{2}\right)}^{2} = \frac{49}{4}$

Add this to both sides of the above equation, giving

${x}^{2} + 7 x + {\left(\frac{7}{2}\right)}^{2} = 9 + {\left(\frac{7}{2}\right)}^{2}$

This creates a perfect square on the left, such that

${\left(x + \frac{7}{2}\right)}^{2} = \frac{85}{4}$

Take the square root of both sides, giving

$x + \frac{7}{2} = \pm \frac{\sqrt{85}}{2}$

Don't forget that $\pm$ in front of your square root operation! Now simply subtract $\frac{7}{2}$ from both sides.

$x = - \frac{7}{2} \pm \frac{\sqrt{85}}{2}$

Plugging that into a calculator gives:

$x \approx - 8.11$ or $x \approx 1.11$