Find All Complex Solutions (x-7)^2=100

${(x - 7)}^{2} = 100$

Step 1

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

Subtract $100$ from both sides of the equation.

${(x - 7)}^{2} - 100 = 0$

Simplify ${(x - 7)}^{2} - 100$ .

Simplify each term.

Rewrite ${(x - 7)}^{2}$ as $(x - 7) (x - 7)$ .

$(x - 7) (x - 7) - 100 = 0$

Expand $(x - 7) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$x (x - 7) - 7 (x - 7) - 100 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 7 - 7 (x - 7) - 100 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7 - 100 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 7 - 7 x - 7 \cdot - 7 - 100 = 0$

Move $- 7$ to the left of $x$ .

$x^{2} - 7 \cdot x - 7 x - 7 \cdot - 7 - 100 = 0$

Multiply $- 7$ by $- 7$ .

$x^{2} - 7 x - 7 x + 49 - 100 = 0$

Subtract $7 x$ from $- 7 x$ .

$x^{2} - 14 x + 49 - 100 = 0$

Subtract $100$ from $49$ .

$x^{2} - 14 x - 51 = 0$

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 $a = 1$ , $b = - 14$ , and $c = - 51$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot - 51)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot - 51}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 51$ .

Multiply $- 4$ by $1$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot - 51}}{2 \cdot 1}$

Multiply $- 4$ by $- 51$ .

$x = \frac{14 \pm \sqrt{196 + 204}}{2 \cdot 1}$

Add $196$ and $204$ .

$x = \frac{14 \pm \sqrt{400}}{2 \cdot 1}$

Rewrite $400$ as $20^{2}$ .

$x = \frac{14 \pm \sqrt{20^{2}}}{2 \cdot 1}$

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

$x = \frac{14 \pm 20}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{14 \pm 20}{2}$

Simplify $\frac{14 \pm 20}{2}$ .

$x = 7 \pm 10$

Step 5

The final answer is the combination of both solutions.

$x = 17, - 3$

Find All Complex Solutions (x-7)^2=100

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