Solve (500-20x)(10+x)=6000

$$( 500 - 20 x ) ( 10 + x ) = 6000$$

Solve for x

$$x=5$$

$$x=10$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 500-20x by 10+x and combine like terms.

$$5000+300x-20x^{2}=6000$$

Subtract 6000 from both sides.

$$5000+300x-20x^{2}-6000=0$$

Subtract 6000 from 5000 to get -1000.

$$-1000+300x-20x^{2}=0$$

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.

$$-20x^{2}+300x-1000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 300 for b, and -1000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-300±\sqrt{300^{2}-4\left(-20\right)\left(-1000\right)}}{2\left(-20\right)}$$

Square 300.

$$x=\frac{-300±\sqrt{90000-4\left(-20\right)\left(-1000\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-300±\sqrt{90000+80\left(-1000\right)}}{2\left(-20\right)}$$

Multiply 80 times -1000.

$$x=\frac{-300±\sqrt{90000-80000}}{2\left(-20\right)}$$

Add 90000 to -80000.

$$x=\frac{-300±\sqrt{10000}}{2\left(-20\right)}$$

Take the square root of 10000.

$$x=\frac{-300±100}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-300±100}{-40}$$

Now solve the equation x=\frac{-300±100}{-40} when ± is plus. Add -300 to 100.

$$x=-\frac{200}{-40}$$

Divide -200 by -40.

$$x=5$$

Now solve the equation x=\frac{-300±100}{-40} when ± is minus. Subtract 100 from -300.

$$x=-\frac{400}{-40}$$

Divide -400 by -40.

$$x=10$$

The equation is now solved.

$$x=5 x=10$$

Steps for Completing the Square

Use the distributive property to multiply 500-20x by 10+x and combine like terms.

$$5000+300x-20x^{2}=6000$$

Subtract 5000 from both sides.

$$300x-20x^{2}=6000-5000$$

Subtract 5000 from 6000 to get 1000.

$$300x-20x^{2}=1000$$

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.

$$-20x^{2}+300x=1000$$

Divide both sides by -20.

$$\frac{-20x^{2}+300x}{-20}=\frac{1000}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{300}{-20}x=\frac{1000}{-20}$$

Divide 300 by -20.

$$x^{2}-15x=\frac{1000}{-20}$$

Divide 1000 by -20.

$$x^{2}-15x=-50$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-50+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-50+\frac{225}{4}$$

Add -50 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{25}{4}$$

Factor x^{2}-15x+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{5}{2} x-\frac{15}{2}=-\frac{5}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=10 x=5$$