# How do you solve 8x^{2} + 21= - 59x?

Mar 14, 2018

$x = - 7 , - 0.375$

#### Explanation:

$8 {x}^{2} + 21 = - 59 x$

add $59 x$ to both sides

$8 {x}^{2} + 59 x + 21$

factor by grouping

$\textcolor{w h i t e}{.11} + 59$
$\textcolor{w h i t e}{.11} \times 168$
. . . . . . . . . .
$\textcolor{w h i t e}{.} \textcolor{w h i t e}{1} 1 \times 168$ $\implies 1 + 168 = 169$
$\textcolor{w h i t e}{.} \textcolor{w h i t e}{1} 2 \times 84 \textcolor{w h i t e}{1}$ $\implies 2 + 84 = 86$
$\textcolor{w h i t e}{.} \textcolor{w h i t e}{1} 3 \times 56 \textcolor{w h i t e}{1}$ $\implies 3 + 56 = \textcolor{red}{59}$

$\left(8 {x}^{2} + 3 x\right) + \left(56 x + 21\right)$
$x \left(8 x + 3\right) + 7 \left(8 x + 3\right)$
$\left(x + 7\right) \left(8 x + 3\right)$
Set each component to $0$ and solve for $x$:

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

Case 1

$8 x + 3 = 0$

$8 x = - 3$

$x = - \frac{3}{8}$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

Case 2

$x + 7 = 0$

$x = - 7$

So $x = - 7$ and $- \frac{3}{8}$ or $- 0.375$

Let's graph the equation and see if we are right

graph{y = 8x^2 + 59x + 21}

We are! Nice job

Mar 14, 2018

$- \frac{3}{8} , - 7$

#### Explanation:

By factoring:
The factors of 8 are (8,1) or (4,2)
The factors of 21 are (21,1) or (7,3)

$21 \cdot 8 = 168$ Too large
$21 \cdot 4 = 84$ Too large
$7 \cdot 8 = 56$ Worth considering

Rearrange formula to equal zero
$8 {x}^{2} + 59 x + 21 = 0$
Since all signs are +, all factors will be +

(8x+?)(x+?)=0

By trial-and-error:
$\left(8 x + 3\right) \left(x + 7\right) = 8 {x}^{2} + 56 x + 3 x + 21 = 8 {x}^{2} + 59 x + 21$

So, $8 x + 3 = 0$ and $x + 7 = 0$ Solve each for $x$

$8 x + 3 = 0$
$8 x = - 3$
$x = - \frac{3}{8}$

And

$x + 7 = 0$
$x = - 7$