Solve Using the Square Root Property x^2+10x+25=11

$x^{2} + 10 x + 25 = 11$

Step 1

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

Subtract $11$ from both sides of the equation.

$x^{2} + 10 x + 25 - 11 = 0$

Subtract $11$ from $25$ .

$x^{2} + 10 x + 14 = 0$

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 $a = 1$ , $b = 10$ , and $c = 14$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (1 \cdot 14)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot 14}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 14$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 14}}{2 \cdot 1}$

Multiply $- 4$ by $14$ .

$x = \frac{- 10 \pm \sqrt{100 - 56}}{2 \cdot 1}$

Subtract $56$ from $100$ .

$x = \frac{- 10 \pm \sqrt{44}}{2 \cdot 1}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{- 10 \pm \sqrt{4 (11)}}{2 \cdot 1}$

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

$x = \frac{- 10 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 10 \pm 2 \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 10 \pm 2 \sqrt{11}}{2}$

Simplify $\frac{- 10 \pm 2 \sqrt{11}}{2}$ .

$x = - 5 \pm \sqrt{11}$

Step 5

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

Simplify the numerator.

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

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot 14}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 14$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 14}}{2 \cdot 1}$

Multiply $- 4$ by $14$ .

$x = \frac{- 10 \pm \sqrt{100 - 56}}{2 \cdot 1}$

Subtract $56$ from $100$ .

$x = \frac{- 10 \pm \sqrt{44}}{2 \cdot 1}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{- 10 \pm \sqrt{4 (11)}}{2 \cdot 1}$

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

$x = \frac{- 10 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 10 \pm 2 \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 10 \pm 2 \sqrt{11}}{2}$

Simplify $\frac{- 10 \pm 2 \sqrt{11}}{2}$ .

$x = - 5 \pm \sqrt{11}$

Change the $\pm$ to $+$ .

$x = - 5 + \sqrt{11}$

Step 6

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

Simplify the numerator.

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

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot 14}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 14$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 14}}{2 \cdot 1}$

Multiply $- 4$ by $14$ .

$x = \frac{- 10 \pm \sqrt{100 - 56}}{2 \cdot 1}$

Subtract $56$ from $100$ .

$x = \frac{- 10 \pm \sqrt{44}}{2 \cdot 1}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{- 10 \pm \sqrt{4 (11)}}{2 \cdot 1}$

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

$x = \frac{- 10 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 10 \pm 2 \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 10 \pm 2 \sqrt{11}}{2}$

Simplify $\frac{- 10 \pm 2 \sqrt{11}}{2}$ .

$x = - 5 \pm \sqrt{11}$

Change the $\pm$ to $-$ .

$x = - 5 - \sqrt{11}$

Step 7

The final answer is the combination of both solutions.

$x = - 5 + \sqrt{11}, - 5 - \sqrt{11}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 5 + \sqrt{11}, - 5 - \sqrt{11}$

Decimal Form:

$x = - 1.68337520 \dots, - 8.31662479 \dots$

Solve Using the Square Root Property x^2+10x+25=11

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