Solve left(x-40right)left(500-10left(x-50right)right)=8000

$$\left( x-40 \right) \left( 500-10 \left( x-50 \right) \right) = 8000$$

Solve for x

$$x=60$$

$$x=80$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -10 by x-50.

$$\left(x-40\right)\left(500-10x+500\right)=8000$$

Add 500 and 500 to get 1000.

$$\left(x-40\right)\left(1000-10x\right)=8000$$

Apply the distributive property by multiplying each term of x-40 by each term of 1000-10x.

$$1000x-10x^{2}-40000+400x=8000$$

Combine 1000x and 400x to get 1400x.

$$1400x-10x^{2}-40000=8000$$

Subtract 8000 from both sides.

$$1400x-10x^{2}-40000-8000=0$$

Subtract 8000 from -40000 to get -48000.

$$1400x-10x^{2}-48000=0$$

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.

$$-10x^{2}+1400x-48000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1400 for b, and -48000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1400±\sqrt{1400^{2}-4\left(-10\right)\left(-48000\right)}}{2\left(-10\right)}$$

Square 1400.

$$x=\frac{-1400±\sqrt{1960000-4\left(-10\right)\left(-48000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1400±\sqrt{1960000+40\left(-48000\right)}}{2\left(-10\right)}$$

Multiply 40 times -48000.

$$x=\frac{-1400±\sqrt{1960000-1920000}}{2\left(-10\right)}$$

Add 1960000 to -1920000.

$$x=\frac{-1400±\sqrt{40000}}{2\left(-10\right)}$$

Take the square root of 40000.

$$x=\frac{-1400±200}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1400±200}{-20}$$

Now solve the equation x=\frac{-1400±200}{-20} when ± is plus. Add -1400 to 200.

$$x=-\frac{1200}{-20}$$

Divide -1200 by -20.

$$x=60$$

Now solve the equation x=\frac{-1400±200}{-20} when ± is minus. Subtract 200 from -1400.

$$x=-\frac{1600}{-20}$$

Divide -1600 by -20.

$$x=80$$

The equation is now solved.

$$x=60 x=80$$

Steps for Completing the Square

Use the distributive property to multiply -10 by x-50.

$$\left(x-40\right)\left(500-10x+500\right)=8000$$

Add 500 and 500 to get 1000.

$$\left(x-40\right)\left(1000-10x\right)=8000$$

Apply the distributive property by multiplying each term of x-40 by each term of 1000-10x.

$$1000x-10x^{2}-40000+400x=8000$$

Combine 1000x and 400x to get 1400x.

$$1400x-10x^{2}-40000=8000$$

Add 40000 to both sides.

$$1400x-10x^{2}=8000+40000$$

Add 8000 and 40000 to get 48000.

$$1400x-10x^{2}=48000$$

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.

$$-10x^{2}+1400x=48000$$

Divide both sides by -10.

$$\frac{-10x^{2}+1400x}{-10}=\frac{48000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{1400}{-10}x=\frac{48000}{-10}$$

Divide 1400 by -10.

$$x^{2}-140x=\frac{48000}{-10}$$

Divide 48000 by -10.

$$x^{2}-140x=-4800$$

Divide -140, the coefficient of the x term, by 2 to get -70. Then add the square of -70 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-140x+\left(-70\right)^{2}=-4800+\left(-70\right)^{2}$$

Square -70.

$$x^{2}-140x+4900=-4800+4900$$

Add -4800 to 4900.

$$x^{2}-140x+4900=100$$

Factor x^{2}-140x+4900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-70\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-70\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-70=10 x-70=-10$$

Add 70 to both sides of the equation.

$$x=80 x=60$$