Solve x*[180-10(x-52)]=2000

$$x \times [ 180 - 10 ( x - 52 ) ] = 2000$$

Solve for x

$$x=5\sqrt{41}+35\approx 67.015621187$$

$$x=35-5\sqrt{41}\approx 2.984378813$$

Steps Using the Quadratic Formula

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

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

Add 180 and 520 to get 700.

$$x\left(700-10x\right)=2000$$

Use the distributive property to multiply x by 700-10x.

$$700x-10x^{2}=2000$$

Subtract 2000 from both sides.

$$700x-10x^{2}-2000=0$$

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.

$$-10x^{2}+700x-2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 700 for b, and -2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-700±\sqrt{700^{2}-4\left(-10\right)\left(-2000\right)}}{2\left(-10\right)}$$

Square 700.

$$x=\frac{-700±\sqrt{490000-4\left(-10\right)\left(-2000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-700±\sqrt{490000+40\left(-2000\right)}}{2\left(-10\right)}$$

Multiply 40 times -2000.

$$x=\frac{-700±\sqrt{490000-80000}}{2\left(-10\right)}$$

Add 490000 to -80000.

$$x=\frac{-700±\sqrt{410000}}{2\left(-10\right)}$$

Take the square root of 410000.

$$x=\frac{-700±100\sqrt{41}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-700±100\sqrt{41}}{-20}$$

Now solve the equation x=\frac{-700±100\sqrt{41}}{-20} when ± is plus. Add -700 to 100\sqrt{41}.

$$x=\frac{100\sqrt{41}-700}{-20}$$

Divide -700+100\sqrt{41} by -20.

$$x=35-5\sqrt{41}$$

Now solve the equation x=\frac{-700±100\sqrt{41}}{-20} when ± is minus. Subtract 100\sqrt{41} from -700.

$$x=\frac{-100\sqrt{41}-700}{-20}$$

Divide -700-100\sqrt{41} by -20.

$$x=5\sqrt{41}+35$$

The equation is now solved.

$$x=35-5\sqrt{41} x=5\sqrt{41}+35$$

Steps for Completing the Square

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

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

Add 180 and 520 to get 700.

$$x\left(700-10x\right)=2000$$

Use the distributive property to multiply x by 700-10x.

$$700x-10x^{2}=2000$$

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.

$$-10x^{2}+700x=2000$$

Divide both sides by -10.

$$\frac{-10x^{2}+700x}{-10}=\frac{2000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{700}{-10}x=\frac{2000}{-10}$$

Divide 700 by -10.

$$x^{2}-70x=\frac{2000}{-10}$$

Divide 2000 by -10.

$$x^{2}-70x=-200$$

Divide -70, the coefficient of the x term, by 2 to get -35. Then add the square of -35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-70x+\left(-35\right)^{2}=-200+\left(-35\right)^{2}$$

Square -35.

$$x^{2}-70x+1225=-200+1225$$

Add -200 to 1225.

$$x^{2}-70x+1225=1025$$

Factor x^{2}-70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-35\right)^{2}=1025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-35\right)^{2}}=\sqrt{1025}$$

Simplify.

$$x-35=5\sqrt{41} x-35=-5\sqrt{41}$$

Add 35 to both sides of the equation.

$$x=5\sqrt{41}+35 x=35-5\sqrt{41}$$