Find All Complex Solutions (x-5)^2-7=8

${(x - 5)}^{2} - 7 = 8$

Step 1

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

Subtract $8$ from both sides of the equation.

${(x - 5)}^{2} - 7 - 8 = 0$

Simplify ${(x - 5)}^{2} - 7 - 8$ .

Simplify each term.

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$(x - 5) (x - 5) - 7 - 8 = 0$

Expand $(x - 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$x (x - 5) - 5 (x - 5) - 7 - 8 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 5 - 5 (x - 5) - 7 - 8 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 7 - 8 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - 7 - 8 = 0$

Move $- 5$ to the left of $x$ .

$x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 - 7 - 8 = 0$

Multiply $- 5$ by $- 5$ .

$x^{2} - 5 x - 5 x + 25 - 7 - 8 = 0$

Subtract $5 x$ from $- 5 x$ .

$x^{2} - 10 x + 25 - 7 - 8 = 0$

Subtract $7$ from $25$ .

$x^{2} - 10 x + 18 - 8 = 0$

Subtract $8$ from $18$ .

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

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 $c = 10$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 1 \cdot 10$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $10$ .

$x = \frac{10 \pm \sqrt{100 - 40}}{2 \cdot 1}$

Subtract $40$ from $100$ .

$x = \frac{10 \pm \sqrt{60}}{2 \cdot 1}$

Rewrite $60$ as $2^{2} \cdot 15$ .

Factor $4$ out of $60$ .

$x = \frac{10 \pm \sqrt{4 (15)}}{2 \cdot 1}$

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{15}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 2 \sqrt{15}}{2}$

Simplify $\frac{10 \pm 2 \sqrt{15}}{2}$ .

$x = 5 \pm \sqrt{15}$

Step 5

The final answer is the combination of both solutions.

$x = 5 + \sqrt{15}, 5 - \sqrt{15}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 5 + \sqrt{15}, 5 - \sqrt{15}$

Decimal Form:

$x = 8.87298334 \dots, 1.12701665 \dots$

Find All Complex Solutions (x-5)^2-7=8

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