Solve (8-x)(8-2x)=16

$$(8-x)(8-2x)=16$$

Solve for x

$$x=2\sqrt{3}+6\approx 9.464101615$$

$$x=6-2\sqrt{3}\approx 2.535898385$$

Steps Using the Quadratic Formula

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

$$64-24x+2x^{2}=16$$

Subtract 16 from both sides.

$$64-24x+2x^{2}-16=0$$

Subtract 16 from 64 to get 48.

$$48-24x+2x^{2}=0$$

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.

$$2x^{2}-24x+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -24 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 2\times 48}}{2\times 2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 2\times 48}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-24\right)±\sqrt{576-8\times 48}}{2\times 2}$$

Multiply -8 times 48.

$$x=\frac{-\left(-24\right)±\sqrt{576-384}}{2\times 2}$$

Add 576 to -384.

$$x=\frac{-\left(-24\right)±\sqrt{192}}{2\times 2}$$

Take the square root of 192.

$$x=\frac{-\left(-24\right)±8\sqrt{3}}{2\times 2}$$

The opposite of -24 is 24.

$$x=\frac{24±8\sqrt{3}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{24±8\sqrt{3}}{4}$$

Now solve the equation x=\frac{24±8\sqrt{3}}{4} when ± is plus. Add 24 to 8\sqrt{3}.

$$x=\frac{8\sqrt{3}+24}{4}$$

Divide 24+8\sqrt{3} by 4.

$$x=2\sqrt{3}+6$$

Now solve the equation x=\frac{24±8\sqrt{3}}{4} when ± is minus. Subtract 8\sqrt{3} from 24.

$$x=\frac{24-8\sqrt{3}}{4}$$

Divide 24-8\sqrt{3} by 4.

$$x=6-2\sqrt{3}$$

The equation is now solved.

$$x=2\sqrt{3}+6 x=6-2\sqrt{3}$$

Steps for Completing the Square

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

$$64-24x+2x^{2}=16$$

Subtract 64 from both sides.

$$-24x+2x^{2}=16-64$$

Subtract 64 from 16 to get -48.

$$-24x+2x^{2}=-48$$

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.

$$2x^{2}-24x=-48$$

Divide both sides by 2.

$$\frac{2x^{2}-24x}{2}=-\frac{48}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{24}{2}\right)x=-\frac{48}{2}$$

Divide -24 by 2.

$$x^{2}-12x=-\frac{48}{2}$$

Divide -48 by 2.

$$x^{2}-12x=-24$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-24+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-24+36$$

Add -24 to 36.

$$x^{2}-12x+36=12$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=12$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{12}$$

Simplify.

$$x-6=2\sqrt{3} x-6=-2\sqrt{3}$$

Add 6 to both sides of the equation.

$$x=2\sqrt{3}+6 x=6-2\sqrt{3}$$