Solve (10+x)(500)(x)=8000

$$( 10 + x ) ( 500 ) ( x ) = 8000$$

Solve for x

$$x=\sqrt{41}-5\approx 1.403124237$$

$$x=-\sqrt{41}-5\approx -11.403124237$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10+x by 500.

$$\left(5000+500x\right)x=8000$$

Use the distributive property to multiply 5000+500x by x.

$$5000x+500x^{2}=8000$$

Subtract 8000 from both sides.

$$5000x+500x^{2}-8000=0$$

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.

$$500x^{2}+5000x-8000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 500 for a, 5000 for b, and -8000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5000±\sqrt{5000^{2}-4\times 500\left(-8000\right)}}{2\times 500}$$

Square 5000.

$$x=\frac{-5000±\sqrt{25000000-4\times 500\left(-8000\right)}}{2\times 500}$$

Multiply -4 times 500.

$$x=\frac{-5000±\sqrt{25000000-2000\left(-8000\right)}}{2\times 500}$$

Multiply -2000 times -8000.

$$x=\frac{-5000±\sqrt{25000000+16000000}}{2\times 500}$$

Add 25000000 to 16000000.

$$x=\frac{-5000±\sqrt{41000000}}{2\times 500}$$

Take the square root of 41000000.

$$x=\frac{-5000±1000\sqrt{41}}{2\times 500}$$

Multiply 2 times 500.

$$x=\frac{-5000±1000\sqrt{41}}{1000}$$

Now solve the equation x=\frac{-5000±1000\sqrt{41}}{1000} when ± is plus. Add -5000 to 1000\sqrt{41}.

$$x=\frac{1000\sqrt{41}-5000}{1000}$$

Divide -5000+1000\sqrt{41} by 1000.

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

Now solve the equation x=\frac{-5000±1000\sqrt{41}}{1000} when ± is minus. Subtract 1000\sqrt{41} from -5000.

$$x=\frac{-1000\sqrt{41}-5000}{1000}$$

Divide -5000-1000\sqrt{41} by 1000.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 10+x by 500.

$$\left(5000+500x\right)x=8000$$

Use the distributive property to multiply 5000+500x by x.

$$5000x+500x^{2}=8000$$

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.

$$500x^{2}+5000x=8000$$

Divide both sides by 500.

$$\frac{500x^{2}+5000x}{500}=\frac{8000}{500}$$

Dividing by 500 undoes the multiplication by 500.

$$x^{2}+\frac{5000}{500}x=\frac{8000}{500}$$

Divide 5000 by 500.

$$x^{2}+10x=\frac{8000}{500}$$

Divide 8000 by 500.

$$x^{2}+10x=16$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=16+5^{2}$$

Square 5.

$$x^{2}+10x+25=16+25$$

Add 16 to 25.

$$x^{2}+10x+25=41$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=41$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{41}$$

Simplify.

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

Subtract 5 from both sides of the equation.

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