Solve 72=2(x+2)^2-8

$$72 = 2 ( x + 2 ) ^ { 2 } - 8$$

Solve for x

$$x=2\sqrt{10}-2\approx 4.32455532$$

$$x=-2\sqrt{10}-2\approx -8.32455532$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$72=2\left(x^{2}+4x+4\right)-8$$

Use the distributive property to multiply 2 by x^{2}+4x+4.

$$72=2x^{2}+8x+8-8$$

Subtract 8 from 8 to get 0.

$$72=2x^{2}+8x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+8x=72$$

Subtract 72 from both sides.

$$2x^{2}+8x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\left(-72\right)}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\left(-72\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\left(-72\right)}}{2\times 2}$$

Multiply -8 times -72.

$$x=\frac{-8±\sqrt{64+576}}{2\times 2}$$

Add 64 to 576.

$$x=\frac{-8±\sqrt{640}}{2\times 2}$$

Take the square root of 640.

$$x=\frac{-8±8\sqrt{10}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±8\sqrt{10}}{4}$$

Now solve the equation x=\frac{-8±8\sqrt{10}}{4} when ± is plus. Add -8 to 8\sqrt{10}.

$$x=\frac{8\sqrt{10}-8}{4}$$

Divide -8+8\sqrt{10} by 4.

$$x=2\sqrt{10}-2$$

Now solve the equation x=\frac{-8±8\sqrt{10}}{4} when ± is minus. Subtract 8\sqrt{10} from -8.

$$x=\frac{-8\sqrt{10}-8}{4}$$

Divide -8-8\sqrt{10} by 4.

$$x=-2\sqrt{10}-2$$

The equation is now solved.

$$x=2\sqrt{10}-2 x=-2\sqrt{10}-2$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$72=2\left(x^{2}+4x+4\right)-8$$

Use the distributive property to multiply 2 by x^{2}+4x+4.

$$72=2x^{2}+8x+8-8$$

Subtract 8 from 8 to get 0.

$$72=2x^{2}+8x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+8x=72$$

Divide both sides by 2.

$$\frac{2x^{2}+8x}{2}=\frac{72}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{8}{2}x=\frac{72}{2}$$

Divide 8 by 2.

$$x^{2}+4x=\frac{72}{2}$$

Divide 72 by 2.

$$x^{2}+4x=36$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=36+2^{2}$$

Square 2.

$$x^{2}+4x+4=36+4$$

Add 36 to 4.

$$x^{2}+4x+4=40$$

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2\right)^{2}=40$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{40}$$

Simplify.

$$x+2=2\sqrt{10} x+2=-2\sqrt{10}$$

Subtract 2 from both sides of the equation.

$$x=2\sqrt{10}-2 x=-2\sqrt{10}-2$$