Solve Using the Square Root Property x^2+4x+4=10

$x^{2} + 4 x + 4 = 10$

Step 1

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

Subtract $10$ from both sides of the equation.

$x^{2} + 4 x + 4 - 10 = 0$

Subtract $10$ from $4$ .

$x^{2} + 4 x - 6 = 0$

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 $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 6)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 6$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4$ by $- 6$ .

$x = \frac{- 4 \pm \sqrt{16 + 24}}{2 \cdot 1}$

Add $16$ and $24$ .

$x = \frac{- 4 \pm \sqrt{40}}{2 \cdot 1}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x = \frac{- 4 \pm \sqrt{4 (10)}}{2 \cdot 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{10}}{2}$

Simplify $\frac{- 4 \pm 2 \sqrt{10}}{2}$ .

$x = - 2 \pm \sqrt{10}$

Step 5

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

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 6$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4$ by $- 6$ .

$x = \frac{- 4 \pm \sqrt{16 + 24}}{2 \cdot 1}$

Add $16$ and $24$ .

$x = \frac{- 4 \pm \sqrt{40}}{2 \cdot 1}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x = \frac{- 4 \pm \sqrt{4 (10)}}{2 \cdot 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{10}}{2}$

Simplify $\frac{- 4 \pm 2 \sqrt{10}}{2}$ .

$x = - 2 \pm \sqrt{10}$

Change the $\pm$ to $+$ .

$x = - 2 + \sqrt{10}$

Step 6

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

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 6$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4$ by $- 6$ .

$x = \frac{- 4 \pm \sqrt{16 + 24}}{2 \cdot 1}$

Add $16$ and $24$ .

$x = \frac{- 4 \pm \sqrt{40}}{2 \cdot 1}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x = \frac{- 4 \pm \sqrt{4 (10)}}{2 \cdot 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{10}}{2}$

Simplify $\frac{- 4 \pm 2 \sqrt{10}}{2}$ .

$x = - 2 \pm \sqrt{10}$

Change the $\pm$ to $-$ .

$x = - 2 - \sqrt{10}$

Step 7

The final answer is the combination of both solutions.

$x = - 2 + \sqrt{10}, - 2 - \sqrt{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 2 + \sqrt{10}, - 2 - \sqrt{10}$

Decimal Form:

$x = 1.16227766 \dots, - 5.16227766 \dots$

Solve Using the Square Root Property x^2+4x+4=10

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