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

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

Solve for x

$$x=460$$

Steps Using the Quadratic Formula

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

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

Add 960 and 160 to get 1120.

$$\left(1120-2x\right)\left(x-360\right)=20000$$

Apply the distributive property by multiplying each term of 1120-2x by each term of x-360.

$$1120x-403200-2x^{2}+720x=20000$$

Combine 1120x and 720x to get 1840x.

$$1840x-403200-2x^{2}=20000$$

Subtract 20000 from both sides.

$$1840x-403200-2x^{2}-20000=0$$

Subtract 20000 from -403200 to get -423200.

$$1840x-423200-2x^{2}=0$$

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.

$$-2x^{2}+1840x-423200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1840 for b, and -423200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1840±\sqrt{1840^{2}-4\left(-2\right)\left(-423200\right)}}{2\left(-2\right)}$$

Square 1840.

$$x=\frac{-1840±\sqrt{3385600-4\left(-2\right)\left(-423200\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-1840±\sqrt{3385600+8\left(-423200\right)}}{2\left(-2\right)}$$

Multiply 8 times -423200.

$$x=\frac{-1840±\sqrt{3385600-3385600}}{2\left(-2\right)}$$

Add 3385600 to -3385600.

$$x=\frac{-1840±\sqrt{0}}{2\left(-2\right)}$$

Take the square root of 0.

$$x=-\frac{1840}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=-\frac{1840}{-4}$$

Divide -1840 by -4.

$$x=460$$

Steps for Completing the Square

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

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

Add 960 and 160 to get 1120.

$$\left(1120-2x\right)\left(x-360\right)=20000$$

Apply the distributive property by multiplying each term of 1120-2x by each term of x-360.

$$1120x-403200-2x^{2}+720x=20000$$

Combine 1120x and 720x to get 1840x.

$$1840x-403200-2x^{2}=20000$$

Add 403200 to both sides.

$$1840x-2x^{2}=20000+403200$$

Add 20000 and 403200 to get 423200.

$$1840x-2x^{2}=423200$$

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.

$$-2x^{2}+1840x=423200$$

Divide both sides by -2.

$$\frac{-2x^{2}+1840x}{-2}=\frac{423200}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{1840}{-2}x=\frac{423200}{-2}$$

Divide 1840 by -2.

$$x^{2}-920x=\frac{423200}{-2}$$

Divide 423200 by -2.

$$x^{2}-920x=-211600$$

Divide -920, the coefficient of the x term, by 2 to get -460. Then add the square of -460 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-920x+\left(-460\right)^{2}=-211600+\left(-460\right)^{2}$$

Square -460.

$$x^{2}-920x+211600=-211600+211600$$

Add -211600 to 211600.

$$x^{2}-920x+211600=0$$

Factor x^{2}-920x+211600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-460\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-460\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-460=0 x-460=0$$

Add 460 to both sides of the equation.

$$x=460 x=460$$

The equation is now solved. Solutions are the same.

$$x=460$$