Solve left(x-360right)left(160+2left(480-xright)right)=2000

$$\left( x-360 \right) \left( 160+2 \left( 480-x \right) \right) = 2000$$

Solve for x

$$x=30\sqrt{10}+460\approx 554.868329805$$

$$x=460-30\sqrt{10}\approx 365.131670195$$

Steps Using the Quadratic Formula

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

$$\left(x-360\right)\left(160+960-2x\right)=2000$$

Add 160 and 960 to get 1120.

$$\left(x-360\right)\left(1120-2x\right)=2000$$

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

$$1120x-2x^{2}-403200+720x=2000$$

Combine 1120x and 720x to get 1840x.

$$1840x-2x^{2}-403200=2000$$

Subtract 2000 from both sides.

$$1840x-2x^{2}-403200-2000=0$$

Subtract 2000 from -403200 to get -405200.

$$1840x-2x^{2}-405200=0$$

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.

$$-2x^{2}+1840x-405200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1840 for b, and -405200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1840±\sqrt{1840^{2}-4\left(-2\right)\left(-405200\right)}}{2\left(-2\right)}$$

Square 1840.

$$x=\frac{-1840±\sqrt{3385600-4\left(-2\right)\left(-405200\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-1840±\sqrt{3385600+8\left(-405200\right)}}{2\left(-2\right)}$$

Multiply 8 times -405200.

$$x=\frac{-1840±\sqrt{3385600-3241600}}{2\left(-2\right)}$$

Add 3385600 to -3241600.

$$x=\frac{-1840±\sqrt{144000}}{2\left(-2\right)}$$

Take the square root of 144000.

$$x=\frac{-1840±120\sqrt{10}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-1840±120\sqrt{10}}{-4}$$

Now solve the equation x=\frac{-1840±120\sqrt{10}}{-4} when ± is plus. Add -1840 to 120\sqrt{10}.

$$x=\frac{120\sqrt{10}-1840}{-4}$$

Divide -1840+120\sqrt{10} by -4.

$$x=460-30\sqrt{10}$$

Now solve the equation x=\frac{-1840±120\sqrt{10}}{-4} when ± is minus. Subtract 120\sqrt{10} from -1840.

$$x=\frac{-120\sqrt{10}-1840}{-4}$$

Divide -1840-120\sqrt{10} by -4.

$$x=30\sqrt{10}+460$$

The equation is now solved.

$$x=460-30\sqrt{10} x=30\sqrt{10}+460$$

Steps for Completing the Square

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

$$\left(x-360\right)\left(160+960-2x\right)=2000$$

Add 160 and 960 to get 1120.

$$\left(x-360\right)\left(1120-2x\right)=2000$$

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

$$1120x-2x^{2}-403200+720x=2000$$

Combine 1120x and 720x to get 1840x.

$$1840x-2x^{2}-403200=2000$$

Add 403200 to both sides.

$$1840x-2x^{2}=2000+403200$$

Add 2000 and 403200 to get 405200.

$$1840x-2x^{2}=405200$$

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.

$$-2x^{2}+1840x=405200$$

Divide both sides by -2.

$$\frac{-2x^{2}+1840x}{-2}=\frac{405200}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{1840}{-2}x=\frac{405200}{-2}$$

Divide 1840 by -2.

$$x^{2}-920x=\frac{405200}{-2}$$

Divide 405200 by -2.

$$x^{2}-920x=-202600$$

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.

$$x^{2}-920x+\left(-460\right)^{2}=-202600+\left(-460\right)^{2}$$

Square -460.

$$x^{2}-920x+211600=-202600+211600$$

Add -202600 to 211600.

$$x^{2}-920x+211600=9000$$

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}.

$$\left(x-460\right)^{2}=9000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-460\right)^{2}}=\sqrt{9000}$$

Simplify.

$$x-460=30\sqrt{10} x-460=-30\sqrt{10}$$

Add 460 to both sides of the equation.

$$x=30\sqrt{10}+460 x=460-30\sqrt{10}$$