Solve 320=2(20+2x)(16+2x)

$$320 = 2 ( 20 + 2 x ) ( 16 + 2 x )$$

Solve for x

$$x=\sqrt{41}-9\approx -2.596875763$$

$$x=-\left(\sqrt{41}+9\right)\approx -15.403124237$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by 20+2x.

$$320=\left(40+4x\right)\left(16+2x\right)$$

Use the distributive property to multiply 40+4x by 16+2x and combine like terms.

$$320=640+144x+8x^{2}$$

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

$$640+144x+8x^{2}=320$$

Subtract 320 from both sides.

$$640+144x+8x^{2}-320=0$$

Subtract 320 from 640 to get 320.

$$320+144x+8x^{2}=0$$

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.

$$8x^{2}+144x+320=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 144 for b, and 320 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-144±\sqrt{144^{2}-4\times 8\times 320}}{2\times 8}$$

Square 144.

$$x=\frac{-144±\sqrt{20736-4\times 8\times 320}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-144±\sqrt{20736-32\times 320}}{2\times 8}$$

Multiply -32 times 320.

$$x=\frac{-144±\sqrt{20736-10240}}{2\times 8}$$

Add 20736 to -10240.

$$x=\frac{-144±\sqrt{10496}}{2\times 8}$$

Take the square root of 10496.

$$x=\frac{-144±16\sqrt{41}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-144±16\sqrt{41}}{16}$$

Now solve the equation x=\frac{-144±16\sqrt{41}}{16} when ± is plus. Add -144 to 16\sqrt{41}.

$$x=\frac{16\sqrt{41}-144}{16}$$

Divide -144+16\sqrt{41} by 16.

$$x=\sqrt{41}-9$$

Now solve the equation x=\frac{-144±16\sqrt{41}}{16} when ± is minus. Subtract 16\sqrt{41} from -144.

$$x=\frac{-16\sqrt{41}-144}{16}$$

Divide -144-16\sqrt{41} by 16.

$$x=-\sqrt{41}-9$$

The equation is now solved.

$$x=\sqrt{41}-9 x=-\sqrt{41}-9$$

Steps for Completing the Square

Use the distributive property to multiply 2 by 20+2x.

$$320=\left(40+4x\right)\left(16+2x\right)$$

Use the distributive property to multiply 40+4x by 16+2x and combine like terms.

$$320=640+144x+8x^{2}$$

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

$$640+144x+8x^{2}=320$$

Subtract 640 from both sides.

$$144x+8x^{2}=320-640$$

Subtract 640 from 320 to get -320.

$$144x+8x^{2}=-320$$

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.

$$8x^{2}+144x=-320$$

Divide both sides by 8.

$$\frac{8x^{2}+144x}{8}=-\frac{320}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{144}{8}x=-\frac{320}{8}$$

Divide 144 by 8.

$$x^{2}+18x=-\frac{320}{8}$$

Divide -320 by 8.

$$x^{2}+18x=-40$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+18x+9^{2}=-40+9^{2}$$

Square 9.

$$x^{2}+18x+81=-40+81$$

Add -40 to 81.

$$x^{2}+18x+81=41$$

Factor x^{2}+18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+9\right)^{2}=41$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{41}$$

Simplify.

$$x+9=\sqrt{41} x+9=-\sqrt{41}$$

Subtract 9 from both sides of the equation.

$$x=\sqrt{41}-9 x=-\sqrt{41}-9$$