Solve Using the Square Root Property 2x^2-20x+50=-128

$2 x^{2} - 20 x + 50 = - 128$

Step 1

Add $128$ to both sides of the equation.

$2 x^{2} - 20 x + 50 + 128 = 0$

Step 2

Add $50$ and $128$ .

$2 x^{2} - 20 x + 178 = 0$

Step 3

Factor $2$ out of $2 x^{2} - 20 x + 178$ .

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) - 20 x + 178 = 0$

Factor $2$ out of $- 20 x$ .

$2 (x^{2}) + 2 (- 10 x) + 178 = 0$

Factor $2$ out of $178$ .

$2 x^{2} + 2 (- 10 x) + 2 \cdot 89 = 0$

Factor $2$ out of $2 x^{2} + 2 (- 10 x)$ .

$2 (x^{2} - 10 x) + 2 \cdot 89 = 0$

Factor $2$ out of $2 (x^{2} - 10 x) + 2 \cdot 89$ .

$2 (x^{2} - 10 x + 89) = 0$

Step 4

Divide each term in $2 (x^{2} - 10 x + 89) = 0$ by $2$ and simplify.

Divide each term in $2 (x^{2} - 10 x + 89) = 0$ by $2$ .

$\frac{2 (x^{2} - 10 x + 89)}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (x^{2} - 10 x + 89)}{2} = \frac{0}{2}$

Divide $x^{2} - 10 x + 89$ by $1$ .

$x^{2} - 10 x + 89 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x^{2} - 10 x + 89 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = - 10$ , and $c = 89$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 89)}}{2 \cdot 1}$

Step 7

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 89}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 89$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 89}}{2 \cdot 1}$

Multiply $- 4$ by $89$ .

$x = \frac{10 \pm \sqrt{100 - 356}}{2 \cdot 1}$

Subtract $356$ from $100$ .

$x = \frac{10 \pm \sqrt{- 256}}{2 \cdot 1}$

Rewrite $- 256$ as $- 1 (256)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 256}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (256)}$ as $\sqrt{- 1} \cdot \sqrt{256}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{256}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{256}}{2 \cdot 1}$

Rewrite $256$ as $16^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{16^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{10 \pm i \cdot 16}{2 \cdot 1}$

Move $16$ to the left of $i$ .

$x = \frac{10 \pm 16 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 16 i}{2}$

Simplify $\frac{10 \pm 16 i}{2}$ .

$x = 5 \pm 8 i$

Step 8

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 89}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 89$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 89}}{2 \cdot 1}$

Multiply $- 4$ by $89$ .

$x = \frac{10 \pm \sqrt{100 - 356}}{2 \cdot 1}$

Subtract $356$ from $100$ .

$x = \frac{10 \pm \sqrt{- 256}}{2 \cdot 1}$

Rewrite $- 256$ as $- 1 (256)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 256}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (256)}$ as $\sqrt{- 1} \cdot \sqrt{256}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{256}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{256}}{2 \cdot 1}$

Rewrite $256$ as $16^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{16^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{10 \pm i \cdot 16}{2 \cdot 1}$

Move $16$ to the left of $i$ .

$x = \frac{10 \pm 16 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 16 i}{2}$

Simplify $\frac{10 \pm 16 i}{2}$ .

$x = 5 \pm 8 i$

Change the $\pm$ to $+$ .

$x = 5 + 8 i$

Step 9

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 89}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 89$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 89}}{2 \cdot 1}$

Multiply $- 4$ by $89$ .

$x = \frac{10 \pm \sqrt{100 - 356}}{2 \cdot 1}$

Subtract $356$ from $100$ .

$x = \frac{10 \pm \sqrt{- 256}}{2 \cdot 1}$

Rewrite $- 256$ as $- 1 (256)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 256}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (256)}$ as $\sqrt{- 1} \cdot \sqrt{256}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{256}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{256}}{2 \cdot 1}$

Rewrite $256$ as $16^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{16^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{10 \pm i \cdot 16}{2 \cdot 1}$

Move $16$ to the left of $i$ .

$x = \frac{10 \pm 16 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 16 i}{2}$

Simplify $\frac{10 \pm 16 i}{2}$ .

$x = 5 \pm 8 i$

Change the $\pm$ to $-$ .

$x = 5 - 8 i$

Step 10

The final answer is the combination of both solutions.

$x = 5 + 8 i, 5 - 8 i$

Solve Using the Square Root Property 2x^2-20x+50=-128

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