Solve (50+x)(500-10x)-40(500-10x)=8000

$$( 50 + x ) ( 500 - 10 x ) - 40 ( 500 - 10 x ) = 8000$$

Solve for x

$$x=10$$

$$x=30$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 50+x by 500-10x and combine like terms.

$$25000-10x^{2}-40\left(500-10x\right)=8000$$

Use the distributive property to multiply -40 by 500-10x.

$$25000-10x^{2}-20000+400x=8000$$

Subtract 20000 from 25000 to get 5000.

$$5000-10x^{2}+400x=8000$$

Subtract 8000 from both sides.

$$5000-10x^{2}+400x-8000=0$$

Subtract 8000 from 5000 to get -3000.

$$-3000-10x^{2}+400x=0$$

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.

$$-10x^{2}+400x-3000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 400 for b, and -3000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-400±\sqrt{400^{2}-4\left(-10\right)\left(-3000\right)}}{2\left(-10\right)}$$

Square 400.

$$x=\frac{-400±\sqrt{160000-4\left(-10\right)\left(-3000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-400±\sqrt{160000+40\left(-3000\right)}}{2\left(-10\right)}$$

Multiply 40 times -3000.

$$x=\frac{-400±\sqrt{160000-120000}}{2\left(-10\right)}$$

Add 160000 to -120000.

$$x=\frac{-400±\sqrt{40000}}{2\left(-10\right)}$$

Take the square root of 40000.

$$x=\frac{-400±200}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-400±200}{-20}$$

Now solve the equation x=\frac{-400±200}{-20} when ± is plus. Add -400 to 200.

$$x=-\frac{200}{-20}$$

Divide -200 by -20.

$$x=10$$

Now solve the equation x=\frac{-400±200}{-20} when ± is minus. Subtract 200 from -400.

$$x=-\frac{600}{-20}$$

Divide -600 by -20.

$$x=30$$

The equation is now solved.

$$x=10 x=30$$

Steps for Completing the Square

Use the distributive property to multiply 50+x by 500-10x and combine like terms.

$$25000-10x^{2}-40\left(500-10x\right)=8000$$

Use the distributive property to multiply -40 by 500-10x.

$$25000-10x^{2}-20000+400x=8000$$

Subtract 20000 from 25000 to get 5000.

$$5000-10x^{2}+400x=8000$$

Subtract 5000 from both sides.

$$-10x^{2}+400x=8000-5000$$

Subtract 5000 from 8000 to get 3000.

$$-10x^{2}+400x=3000$$

Divide both sides by -10.

$$\frac{-10x^{2}+400x}{-10}=\frac{3000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{400}{-10}x=\frac{3000}{-10}$$

Divide 400 by -10.

$$x^{2}-40x=\frac{3000}{-10}$$

Divide 3000 by -10.

$$x^{2}-40x=-300$$

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.

$$x^{2}-40x+\left(-20\right)^{2}=-300+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-300+400$$

Add -300 to 400.

$$x^{2}-40x+400=100$$

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}.

$$\left(x-20\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-20=10 x-20=-10$$

Add 20 to both sides of the equation.

$$x=30 x=10$$