Solve (x-30)(600-10x)=2000

$$( x - 30 ) ( 600 - 10 x ) = 2000$$

Solve for x

$$x=40$$

$$x=50$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-30 by 600-10x and combine like terms.

$$900x-10x^{2}-18000=2000$$

Subtract 2000 from both sides.

$$900x-10x^{2}-18000-2000=0$$

Subtract 2000 from -18000 to get -20000.

$$900x-10x^{2}-20000=0$$

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.

$$-10x^{2}+900x-20000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 900 for b, and -20000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-900±\sqrt{900^{2}-4\left(-10\right)\left(-20000\right)}}{2\left(-10\right)}$$

Square 900.

$$x=\frac{-900±\sqrt{810000-4\left(-10\right)\left(-20000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-900±\sqrt{810000+40\left(-20000\right)}}{2\left(-10\right)}$$

Multiply 40 times -20000.

$$x=\frac{-900±\sqrt{810000-800000}}{2\left(-10\right)}$$

Add 810000 to -800000.

$$x=\frac{-900±\sqrt{10000}}{2\left(-10\right)}$$

Take the square root of 10000.

$$x=\frac{-900±100}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-900±100}{-20}$$

Now solve the equation x=\frac{-900±100}{-20} when ± is plus. Add -900 to 100.

$$x=-\frac{800}{-20}$$

Divide -800 by -20.

$$x=40$$

Now solve the equation x=\frac{-900±100}{-20} when ± is minus. Subtract 100 from -900.

$$x=-\frac{1000}{-20}$$

Divide -1000 by -20.

$$x=50$$

The equation is now solved.

$$x=40 x=50$$

Steps for Completing the Square

Use the distributive property to multiply x-30 by 600-10x and combine like terms.

$$900x-10x^{2}-18000=2000$$

Add 18000 to both sides.

$$900x-10x^{2}=2000+18000$$

Add 2000 and 18000 to get 20000.

$$900x-10x^{2}=20000$$

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.

$$-10x^{2}+900x=20000$$

Divide both sides by -10.

$$\frac{-10x^{2}+900x}{-10}=\frac{20000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{900}{-10}x=\frac{20000}{-10}$$

Divide 900 by -10.

$$x^{2}-90x=\frac{20000}{-10}$$

Divide 20000 by -10.

$$x^{2}-90x=-2000$$

Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-90x+\left(-45\right)^{2}=-2000+\left(-45\right)^{2}$$

Square -45.

$$x^{2}-90x+2025=-2000+2025$$

Add -2000 to 2025.

$$x^{2}-90x+2025=25$$

Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-45\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-45\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-45=5 x-45=-5$$

Add 45 to both sides of the equation.

$$x=50 x=40$$