Solve by Factoring x^2+24x+100=-46

$x^{2} + 24 x + 100 = - 46$

Step 1

Add $46$ to both sides of the equation.

$x^{2} + 24 x + 100 + 46 = 0$

Step 2

Add $100$ and $46$ .

$x^{2} + 24 x + 146 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = 24$ , and $c = 146$ into the quadratic formula and solve for $x$ .

$\frac{- 24 \pm \sqrt{24^{2} - 4 \cdot (1 \cdot 146)}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 1 \cdot 146}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 146$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 146}}{2 \cdot 1}$

Multiply $- 4$ by $146$ .

$x = \frac{- 24 \pm \sqrt{576 - 584}}{2 \cdot 1}$

Subtract $584$ from $576$ .

$x = \frac{- 24 \pm \sqrt{- 8}}{2 \cdot 1}$

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 24 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 24 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 24 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$x = \frac{- 24 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 24 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 24 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{- 24 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 24 \pm 2 i \sqrt{2}}{2}$

Simplify $\frac{- 24 \pm 2 i \sqrt{2}}{2}$ .

$x = - 12 \pm i \sqrt{2}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 1 \cdot 146}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 146$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 146}}{2 \cdot 1}$

Multiply $- 4$ by $146$ .

$x = \frac{- 24 \pm \sqrt{576 - 584}}{2 \cdot 1}$

Subtract $584$ from $576$ .

$x = \frac{- 24 \pm \sqrt{- 8}}{2 \cdot 1}$

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 24 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 24 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 24 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$x = \frac{- 24 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 24 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 24 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{- 24 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 24 \pm 2 i \sqrt{2}}{2}$

Simplify $\frac{- 24 \pm 2 i \sqrt{2}}{2}$ .

$x = - 12 \pm i \sqrt{2}$

Change the $\pm$ to $+$ .

$x = - 12 + i \sqrt{2}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 1 \cdot 146}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 146$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 146}}{2 \cdot 1}$

Multiply $- 4$ by $146$ .

$x = \frac{- 24 \pm \sqrt{576 - 584}}{2 \cdot 1}$

Subtract $584$ from $576$ .

$x = \frac{- 24 \pm \sqrt{- 8}}{2 \cdot 1}$

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 24 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 24 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 24 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$x = \frac{- 24 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 24 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 24 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{- 24 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 24 \pm 2 i \sqrt{2}}{2}$

Simplify $\frac{- 24 \pm 2 i \sqrt{2}}{2}$ .

$x = - 12 \pm i \sqrt{2}$

Change the $\pm$ to $-$ .

$x = - 12 - i \sqrt{2}$

Step 8

The final answer is the combination of both solutions.

$x = - 12 + i \sqrt{2}, - 12 - i \sqrt{2}$

Solve by Factoring x^2+24x+100=-46

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