Solve Using the Square Root Property x^2-4x+4=-25

$x^{2} - 4 x + 4 = - 25$

Step 1

Move all terms to the left side of the equation and simplify.

Add $25$ to both sides of the equation.

$x^{2} - 4 x + 4 + 25 = 0$

Add $4$ and $25$ .

$x^{2} - 4 x + 29 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - 4$ , and $c = 29$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 29)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 29$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 29}}{2 \cdot 1}$

Multiply $- 4$ by $29$ .

$x = \frac{4 \pm \sqrt{16 - 116}}{2 \cdot 1}$

Subtract $116$ from $16$ .

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

Rewrite $- 100$ as $- 1 (100)$ .

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

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

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

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

$x = \frac{4 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot 10}{2 \cdot 1}$

Move $10$ to the left of $i$ .

$x = \frac{4 \pm 10 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 10 i}{2}$

Simplify $\frac{4 \pm 10 i}{2}$ .

$x = 2 \pm 5 i$

Step 5

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 29$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 29}}{2 \cdot 1}$

Multiply $- 4$ by $29$ .

$x = \frac{4 \pm \sqrt{16 - 116}}{2 \cdot 1}$

Subtract $116$ from $16$ .

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

Rewrite $- 100$ as $- 1 (100)$ .

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

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

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

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

$x = \frac{4 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot 10}{2 \cdot 1}$

Move $10$ to the left of $i$ .

$x = \frac{4 \pm 10 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 10 i}{2}$

Simplify $\frac{4 \pm 10 i}{2}$ .

$x = 2 \pm 5 i$

Change the $\pm$ to $+$ .

$x = 2 + 5 i$

Step 6

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 29$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 29}}{2 \cdot 1}$

Multiply $- 4$ by $29$ .

$x = \frac{4 \pm \sqrt{16 - 116}}{2 \cdot 1}$

Subtract $116$ from $16$ .

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

Rewrite $- 100$ as $- 1 (100)$ .

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

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

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

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

$x = \frac{4 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot 10}{2 \cdot 1}$

Move $10$ to the left of $i$ .

$x = \frac{4 \pm 10 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 10 i}{2}$

Simplify $\frac{4 \pm 10 i}{2}$ .

$x = 2 \pm 5 i$

Change the $\pm$ to $-$ .

$x = 2 - 5 i$

Step 7

The final answer is the combination of both solutions.

$x = 2 + 5 i, 2 - 5 i$

Solve Using the Square Root Property x^2-4x+4=-25

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