Solve 20000=(x-360)(160-2(480-x)

$$20000 = ( x - 360 ) ( 160 - 2 ( 480 - x )$$

Solve for x

$$x=20\sqrt{26}+380\approx 481.980390272$$

$$x=380-20\sqrt{26}\approx 278.019609728$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -2 by 480-x.

$$20000=\left(x-360\right)\left(160-960+2x\right)$$

Subtract 960 from 160 to get -800.

$$20000=\left(x-360\right)\left(-800+2x\right)$$

Apply the distributive property by multiplying each term of x-360 by each term of -800+2x.

$$20000=-800x+2x^{2}[phone]x$$

Combine -800x and -720x to get -1520x.

$$20000=-1520x+2x^{2}+288000$$

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

$$-1520x+2x^{2}+288000=20000$$

Subtract 20000 from both sides.

$$-1520x+2x^{2}[phone]=0$$

Subtract 20000 from 288000 to get 268000.

$$-1520x+2x^{2}+268000=0$$

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.

$$2x^{2}-1520x+268000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -1520 for b, and 268000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1520\right)±\sqrt{\left(-1520\right)^{2}-4\times 2\times 268000}}{2\times 2}$$

Square -1520.

$$x=\frac{-\left(-1520\right)±\sqrt{2310400-4\times 2\times 268000}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-1520\right)±\sqrt{2310400-8\times 268000}}{2\times 2}$$

Multiply -8 times 268000.

$$x=\frac{-\left(-1520\right)±\sqrt{2310400-2144000}}{2\times 2}$$

Add 2310400 to -2144000.

$$x=\frac{-\left(-1520\right)±\sqrt{166400}}{2\times 2}$$

Take the square root of 166400.

$$x=\frac{-\left(-1520\right)±80\sqrt{26}}{2\times 2}$$

The opposite of -1520 is 1520.

$$x=\frac{1520±80\sqrt{26}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{1520±80\sqrt{26}}{4}$$

Now solve the equation x=\frac{1520±80\sqrt{26}}{4} when ± is plus. Add 1520 to 80\sqrt{26}.

$$x=\frac{80\sqrt{26}+1520}{4}$$

Divide 1520+80\sqrt{26} by 4.

$$x=20\sqrt{26}+380$$

Now solve the equation x=\frac{1520±80\sqrt{26}}{4} when ± is minus. Subtract 80\sqrt{26} from 1520.

$$x=\frac{1520-80\sqrt{26}}{4}$$

Divide 1520-80\sqrt{26} by 4.

$$x=380-20\sqrt{26}$$

The equation is now solved.

$$x=20\sqrt{26}+380 x=380-20\sqrt{26}$$

Steps for Completing the Square

Use the distributive property to multiply -2 by 480-x.

$$20000=\left(x-360\right)\left(160-960+2x\right)$$

Subtract 960 from 160 to get -800.

$$20000=\left(x-360\right)\left(-800+2x\right)$$

Apply the distributive property by multiplying each term of x-360 by each term of -800+2x.

$$20000=-800x+2x^{2}[phone]x$$

Combine -800x and -720x to get -1520x.

$$20000=-1520x+2x^{2}+288000$$

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

$$-1520x+2x^{2}+288000=20000$$

Subtract 288000 from both sides.

$$-1520x+2x^{2}=20000-288000$$

Subtract 288000 from 20000 to get -268000.

$$-1520x+2x^{2}=-268000$$

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.

$$2x^{2}-1520x=-268000$$

Divide both sides by 2.

$$\frac{2x^{2}-1520x}{2}=-\frac{268000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{1520}{2}\right)x=-\frac{268000}{2}$$

Divide -1520 by 2.

$$x^{2}-760x=-\frac{268000}{2}$$

Divide -268000 by 2.

$$x^{2}-760x=-134000$$

Divide -760, the coefficient of the x term, by 2 to get -380. Then add the square of -380 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-760x+\left(-380\right)^{2}=-134000+\left(-380\right)^{2}$$

Square -380.

$$x^{2}-760x+144400=-134000+144400$$

Add -134000 to 144400.

$$x^{2}-760x+144400=10400$$

Factor x^{2}-760x+144400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-380\right)^{2}=10400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-380\right)^{2}}=\sqrt{10400}$$

Simplify.

$$x-380=20\sqrt{26} x-380=-20\sqrt{26}$$

Add 380 to both sides of the equation.

$$x=20\sqrt{26}+380 x=380-20\sqrt{26}$$