Solve Using the Quadratic Formula 2(x-7)^2=2

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

Step 1

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

Subtract $2$ from both sides of the equation.

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

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

Simplify each term.

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

$2 ((x - 7) (x - 7)) - 2 = 0$

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

Apply the distributive property.

$2 (x (x - 7) - 7 (x - 7)) - 2 = 0$

Apply the distributive property.

$2 (x \cdot x + x \cdot - 7 - 7 (x - 7)) - 2 = 0$

Apply the distributive property.

$2 (x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7) - 2 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 7$ by $- 7$ .

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

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

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

Apply the distributive property.

$2 x^{2} + 2 (- 14 x) + 2 \cdot 49 - 2 = 0$

Simplify.

Multiply $- 14$ by $2$ .

$2 x^{2} - 28 x + 2 \cdot 49 - 2 = 0$

Multiply $2$ by $49$ .

$2 x^{2} - 28 x + 98 - 2 = 0$

Subtract $2$ from $98$ .

$2 x^{2} - 28 x + 96 = 0$

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 $a = 2$ , $b = - 28$ , and $c = 96$ into the quadratic formula and solve for $x$ .

$\frac{28 \pm \sqrt{{(- 28)}^{2} - 4 \cdot (2 \cdot 96)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 28$ to the power of $2$ .

$x = \frac{28 \pm \sqrt{784 - 4 \cdot 2 \cdot 96}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 96$ .

Multiply $- 4$ by $2$ .

$x = \frac{28 \pm \sqrt{784 - 8 \cdot 96}}{2 \cdot 2}$

Multiply $- 8$ by $96$ .

$x = \frac{28 \pm \sqrt{784 - 768}}{2 \cdot 2}$

Subtract $768$ from $784$ .

$x = \frac{28 \pm \sqrt{16}}{2 \cdot 2}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{28 \pm \sqrt{4^{2}}}{2 \cdot 2}$

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

$x = \frac{28 \pm 4}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{28 \pm 4}{4}$

Simplify $\frac{28 \pm 4}{4}$ .

$x = 7 \pm 1$

Step 5

The final answer is the combination of both solutions.

$x = 8, 6$

Solve Using the Quadratic Formula 2(x-7)^2=2

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