Solve (80-4x)(8-2x)=1080

$$( 80 - 4 x ) ( 8 - 2 x ) = 1080$$

Solve for x

$$x=\sqrt{199}+12\approx 26.10673598$$

$$x=12-\sqrt{199}\approx -2.10673598$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 80-4x by 8-2x and combine like terms.

$$640-192x+8x^{2}=1080$$

Subtract 1080 from both sides.

$$640-192x+8x^{2}-1080=0$$

Subtract 1080 from 640 to get -440.

$$-440-192x+8x^{2}=0$$

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.

$$8x^{2}-192x-440=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -192 for b, and -440 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-192\right)±\sqrt{\left(-192\right)^{2}-4\times 8\left(-440\right)}}{2\times 8}$$

Square -192.

$$x=\frac{-\left(-192\right)±\sqrt{36864-4\times 8\left(-440\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-192\right)±\sqrt{36864-32\left(-440\right)}}{2\times 8}$$

Multiply -32 times -440.

$$x=\frac{-\left(-192\right)±\sqrt{36864+14080}}{2\times 8}$$

Add 36864 to 14080.

$$x=\frac{-\left(-192\right)±\sqrt{50944}}{2\times 8}$$

Take the square root of 50944.

$$x=\frac{-\left(-192\right)±16\sqrt{199}}{2\times 8}$$

The opposite of -192 is 192.

$$x=\frac{192±16\sqrt{199}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{192±16\sqrt{199}}{16}$$

Now solve the equation x=\frac{192±16\sqrt{199}}{16} when ± is plus. Add 192 to 16\sqrt{199}.

$$x=\frac{16\sqrt{199}+192}{16}$$

Divide 192+16\sqrt{199} by 16.

$$x=\sqrt{199}+12$$

Now solve the equation x=\frac{192±16\sqrt{199}}{16} when ± is minus. Subtract 16\sqrt{199} from 192.

$$x=\frac{192-16\sqrt{199}}{16}$$

Divide 192-16\sqrt{199} by 16.

$$x=12-\sqrt{199}$$

The equation is now solved.

$$x=\sqrt{199}+12 x=12-\sqrt{199}$$

Steps for Completing the Square

Use the distributive property to multiply 80-4x by 8-2x and combine like terms.

$$640-192x+8x^{2}=1080$$

Subtract 640 from both sides.

$$-192x+8x^{2}=1080-640$$

Subtract 640 from 1080 to get 440.

$$-192x+8x^{2}=440$$

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.

$$8x^{2}-192x=440$$

Divide both sides by 8.

$$\frac{8x^{2}-192x}{8}=\frac{440}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{192}{8}\right)x=\frac{440}{8}$$

Divide -192 by 8.

$$x^{2}-24x=\frac{440}{8}$$

Divide 440 by 8.

$$x^{2}-24x=55$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=55+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=55+144$$

Add 55 to 144.

$$x^{2}-24x+144=199$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=199$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{199}$$

Simplify.

$$x-12=\sqrt{199} x-12=-\sqrt{199}$$

Add 12 to both sides of the equation.

$$x=\sqrt{199}+12 x=12-\sqrt{199}$$