Solve by Factoring (x-4)^2+4=49

${(x - 4)}^{2} + 4 = 49$

Step 1

Subtract $49$ from both sides of the equation.

${(x - 4)}^{2} + 4 - 49 = 0$

Step 2

Simplify ${(x - 4)}^{2} + 4 - 49$ .

Simplify each term.

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$(x - 4) (x - 4) + 4 - 49 = 0$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$x (x - 4) - 4 (x - 4) + 4 - 49 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 4 - 4 (x - 4) + 4 - 49 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 + 4 - 49 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4 + 4 - 49 = 0$

Move $- 4$ to the left of $x$ .

$x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4 + 4 - 49 = 0$

Multiply $- 4$ by $- 4$ .

$x^{2} - 4 x - 4 x + 16 + 4 - 49 = 0$

Subtract $4 x$ from $- 4 x$ .

$x^{2} - 8 x + 16 + 4 - 49 = 0$

Add $16$ and $4$ .

$x^{2} - 8 x + 20 - 49 = 0$

Subtract $49$ from $20$ .

$x^{2} - 8 x - 29 = 0$

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 $a = 1$ , $b = - 8$ , and $c = - 29$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot - 29)}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 29}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 29}}{2 \cdot 1}$

Multiply $- 4$ by $- 29$ .

$x = \frac{8 \pm \sqrt{64 + 116}}{2 \cdot 1}$

Add $64$ and $116$ .

$x = \frac{8 \pm \sqrt{180}}{2 \cdot 1}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$x = \frac{8 \pm \sqrt{36 (5)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{8 \pm \sqrt{6^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{8 \pm 6 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 6 \sqrt{5}}{2}$

Simplify $\frac{8 \pm 6 \sqrt{5}}{2}$ .

$x = 4 \pm 3 \sqrt{5}$

Step 6

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

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 29}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 29}}{2 \cdot 1}$

Multiply $- 4$ by $- 29$ .

$x = \frac{8 \pm \sqrt{64 + 116}}{2 \cdot 1}$

Add $64$ and $116$ .

$x = \frac{8 \pm \sqrt{180}}{2 \cdot 1}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$x = \frac{8 \pm \sqrt{36 (5)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{8 \pm \sqrt{6^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{8 \pm 6 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 6 \sqrt{5}}{2}$

Simplify $\frac{8 \pm 6 \sqrt{5}}{2}$ .

$x = 4 \pm 3 \sqrt{5}$

Change the $\pm$ to $+$ .

$x = 4 + 3 \sqrt{5}$

Step 7

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

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 29}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 29}}{2 \cdot 1}$

Multiply $- 4$ by $- 29$ .

$x = \frac{8 \pm \sqrt{64 + 116}}{2 \cdot 1}$

Add $64$ and $116$ .

$x = \frac{8 \pm \sqrt{180}}{2 \cdot 1}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$x = \frac{8 \pm \sqrt{36 (5)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{8 \pm \sqrt{6^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{8 \pm 6 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 6 \sqrt{5}}{2}$

Simplify $\frac{8 \pm 6 \sqrt{5}}{2}$ .

$x = 4 \pm 3 \sqrt{5}$

Change the $\pm$ to $-$ .

$x = 4 - 3 \sqrt{5}$

Step 8

The final answer is the combination of both solutions.

$x = 4 + 3 \sqrt{5}, 4 - 3 \sqrt{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 4 + 3 \sqrt{5}, 4 - 3 \sqrt{5}$

Decimal Form:

$x = 10.70820393 \dots, - 2.70820393 \dots$

Solve by Factoring (x-4)^2+4=49

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