# Solve the equation by completing the square. 8x2= -11x-7 ?

Jun 24, 2018

$x = - \frac{11}{16} \pm \frac{\sqrt{103}}{4} i = - \frac{1}{16} \left(11 \pm 4 \sqrt{103} i\right)$

#### Explanation:

As 8x2 can be read as 8 times 2, I would advice you to write this as 8x^2 to ensure you are not misunderstood. This is $8 {x}^{2}$

It's useful to start with drawing a graph:

As the graph does not cross the x axis, this means the solutions are complex, something which is useful to know before we start.

As we want to complete the square, we write the expression as
$8 {x}^{2} + 11 x = - 11 x - 7 + 11 x = - 7$
Divide all therms with 8:
${x}^{2} + \frac{11}{8} x = - \frac{7}{8}$

We want to write the left side on the form
${\left(x + a\right)}^{2} = {x}^{2} + 2 a x + {a}^{2}$
Therefore $2 a = \frac{11}{8}$ or $a = \frac{11}{16}$
Add ${\left(\frac{11}{16}\right)}^{2} = \frac{121}{16} ^ 2$ to both sides to fulfill the square:
${x}^{2} + 2 \cdot \frac{11}{16} x + {\left(\frac{11}{16}\right)}^{2} = - \frac{7}{8} + \frac{121}{16} ^ 2$
=$\frac{- 7 \cdot 32 + 121}{16} ^ 2 = \frac{- 103}{16} ^ 2$
${\left(x + \frac{11}{16}\right)}^{2} = \frac{- 103}{16} ^ 2$

Therefore:
$x + \frac{11}{16} = \frac{\sqrt{103}}{4} i$
$x = - \frac{1}{16} \left(11 \pm 4 \sqrt{103} i\right)$