Solve (4+2x)(8+2x)=64

$$( 4 + 2 x ) ( 8 + 2 x ) = 64$$

Solve for x

$$x=\sqrt{17}-3\approx 1.123105626$$

$$x=-\left(\sqrt{17}+3\right)\approx -7.123105626$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4+2x by 8+2x and combine like terms.

$$32+24x+4x^{2}=64$$

Subtract 64 from both sides.

$$32+24x+4x^{2}-64=0$$

Subtract 64 from 32 to get -32.

$$-32+24x+4x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$4x^{2}+24x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 24 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 4\left(-32\right)}}{2\times 4}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 4\left(-32\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-24±\sqrt{576-16\left(-32\right)}}{2\times 4}$$

Multiply -16 times -32.

$$x=\frac{-24±\sqrt{576+512}}{2\times 4}$$

Add 576 to 512.

$$x=\frac{-24±\sqrt{1088}}{2\times 4}$$

Take the square root of 1088.

$$x=\frac{-24±8\sqrt{17}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-24±8\sqrt{17}}{8}$$

Now solve the equation x=\frac{-24±8\sqrt{17}}{8} when ± is plus. Add -24 to 8\sqrt{17}.

$$x=\frac{8\sqrt{17}-24}{8}$$

Divide -24+8\sqrt{17} by 8.

$$x=\sqrt{17}-3$$

Now solve the equation x=\frac{-24±8\sqrt{17}}{8} when ± is minus. Subtract 8\sqrt{17} from -24.

$$x=\frac{-8\sqrt{17}-24}{8}$$

Divide -24-8\sqrt{17} by 8.

$$x=-\sqrt{17}-3$$

The equation is now solved.

$$x=\sqrt{17}-3 x=-\sqrt{17}-3$$

Steps for Completing the Square

Use the distributive property to multiply 4+2x by 8+2x and combine like terms.

$$32+24x+4x^{2}=64$$

Subtract 32 from both sides.

$$24x+4x^{2}=64-32$$

Subtract 32 from 64 to get 32.

$$24x+4x^{2}=32$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$4x^{2}+24x=32$$

Divide both sides by 4.

$$\frac{4x^{2}+24x}{4}=\frac{32}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{24}{4}x=\frac{32}{4}$$

Divide 24 by 4.

$$x^{2}+6x=\frac{32}{4}$$

Divide 32 by 4.

$$x^{2}+6x=8$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=8+3^{2}$$

Square 3.

$$x^{2}+6x+9=8+9$$

Add 8 to 9.

$$x^{2}+6x+9=17$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=17$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{17}$$

Simplify.

$$x+3=\sqrt{17} x+3=-\sqrt{17}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{17}-3 x=-\sqrt{17}-3$$