Solve (x-(6))(180-10(52-x)]=2000

$$( x - ( 6 ) ) ( 180 - 10 ( 52 - x ) ] = 2000$$

Solve for x

$$x=6\sqrt{11}+20\approx 39.899748742$$

$$x=20-6\sqrt{11}\approx 0.100251258$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -10 by 52-x.

$$\left(x-6\right)\left(180-520+10x\right)=2000$$

Subtract 520 from 180 to get -340.

$$\left(x-6\right)\left(-340+10x\right)=2000$$

Apply the distributive property by multiplying each term of x-6 by each term of -340+10x.

$$-340x+10x^{2}+2040-60x=2000$$

Combine -340x and -60x to get -400x.

$$-400x+10x^{2}+2040=2000$$

Subtract 2000 from both sides.

$$-400x+10x^{2}[phone]=0$$

Subtract 2000 from 2040 to get 40.

$$-400x+10x^{2}+40=0$$

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.

$$10x^{2}-400x+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -400 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-400\right)±\sqrt{\left(-400\right)^{2}-4\times 10\times 40}}{2\times 10}$$

Square -400.

$$x=\frac{-\left(-400\right)±\sqrt{160000-4\times 10\times 40}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-400\right)±\sqrt{160000-40\times 40}}{2\times 10}$$

Multiply -40 times 40.

$$x=\frac{-\left(-400\right)±\sqrt{160000-1600}}{2\times 10}$$

Add 160000 to -1600.

$$x=\frac{-\left(-400\right)±\sqrt{158400}}{2\times 10}$$

Take the square root of 158400.

$$x=\frac{-\left(-400\right)±120\sqrt{11}}{2\times 10}$$

The opposite of -400 is 400.

$$x=\frac{400±120\sqrt{11}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{400±120\sqrt{11}}{20}$$

Now solve the equation x=\frac{400±120\sqrt{11}}{20} when ± is plus. Add 400 to 120\sqrt{11}.

$$x=\frac{120\sqrt{11}+400}{20}$$

Divide 400+120\sqrt{11} by 20.

$$x=6\sqrt{11}+20$$

Now solve the equation x=\frac{400±120\sqrt{11}}{20} when ± is minus. Subtract 120\sqrt{11} from 400.

$$x=\frac{400-120\sqrt{11}}{20}$$

Divide 400-120\sqrt{11} by 20.

$$x=20-6\sqrt{11}$$

The equation is now solved.

$$x=6\sqrt{11}+20 x=20-6\sqrt{11}$$

Steps for Completing the Square

Use the distributive property to multiply -10 by 52-x.

$$\left(x-6\right)\left(180-520+10x\right)=2000$$

Subtract 520 from 180 to get -340.

$$\left(x-6\right)\left(-340+10x\right)=2000$$

Apply the distributive property by multiplying each term of x-6 by each term of -340+10x.

$$-340x+10x^{2}+2040-60x=2000$$

Combine -340x and -60x to get -400x.

$$-400x+10x^{2}+2040=2000$$

Subtract 2040 from both sides.

$$-400x+10x^{2}=2000-2040$$

Subtract 2040 from 2000 to get -40.

$$-400x+10x^{2}=-40$$

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.

$$10x^{2}-400x=-40$$

Divide both sides by 10.

$$\frac{10x^{2}-400x}{10}=-\frac{40}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{400}{10}\right)x=-\frac{40}{10}$$

Divide -400 by 10.

$$x^{2}-40x=-\frac{40}{10}$$

Divide -40 by 10.

$$x^{2}-40x=-4$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=-4+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-4+400$$

Add -4 to 400.

$$x^{2}-40x+400=396$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=396$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{396}$$

Simplify.

$$x-20=6\sqrt{11} x-20=-6\sqrt{11}$$

Add 20 to both sides of the equation.

$$x=6\sqrt{11}+20 x=20-6\sqrt{11}$$