Solve 3010=(20-x)(300+20x)

$$3010 = ( 20 - x ) ( 300 + 20 x )$$

Solve for x

$$x = \frac{\sqrt{623} + 5}{2} \approx 14.979983974$$

$$x=\frac{5-\sqrt{623}}{2}\approx -9.979983974$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-x by 300+20x and combine like terms.

$$3010=6000+100x-20x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$6000+100x-20x^{2}=3010$$

Subtract 3010 from both sides.

$$6000+100x-20x^{2}-3010=0$$

Subtract 3010 from 6000 to get 2990.

$$2990+100x-20x^{2}=0$$

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.

$$-20x^{2}+100x+2990=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 100 for b, and 2990 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-20\right)\times 2990}}{2\left(-20\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-20\right)\times 2990}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-100±\sqrt{10000+80\times 2990}}{2\left(-20\right)}$$

Multiply 80 times 2990.

$$x=\frac{-100±\sqrt{10000+239200}}{2\left(-20\right)}$$

Add 10000 to 239200.

$$x=\frac{-100±\sqrt{249200}}{2\left(-20\right)}$$

Take the square root of 249200.

$$x=\frac{-100±20\sqrt{623}}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-100±20\sqrt{623}}{-40}$$

Now solve the equation x=\frac{-100±20\sqrt{623}}{-40} when ± is plus. Add -100 to 20\sqrt{623}.

$$x=\frac{20\sqrt{623}-100}{-40}$$

Divide -100+20\sqrt{623} by -40.

$$x=\frac{5-\sqrt{623}}{2}$$

Now solve the equation x=\frac{-100±20\sqrt{623}}{-40} when ± is minus. Subtract 20\sqrt{623} from -100.

$$x=\frac{-20\sqrt{623}-100}{-40}$$

Divide -100-20\sqrt{623} by -40.

$$x=\frac{\sqrt{623}+5}{2}$$

The equation is now solved.

$$x=\frac{5-\sqrt{623}}{2} x=\frac{\sqrt{623}+5}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 20-x by 300+20x and combine like terms.

$$3010=6000+100x-20x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$6000+100x-20x^{2}=3010$$

Subtract 6000 from both sides.

$$100x-20x^{2}=3010-6000$$

Subtract 6000 from 3010 to get -2990.

$$100x-20x^{2}=-2990$$

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.

$$-20x^{2}+100x=-2990$$

Divide both sides by -20.

$$\frac{-20x^{2}+100x}{-20}=-\frac{2990}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{100}{-20}x=-\frac{2990}{-20}$$

Divide 100 by -20.

$$x^{2}-5x=-\frac{2990}{-20}$$

Reduce the fraction \frac{-2990}{-20} to lowest terms by extracting and canceling out 10.

$$x^{2}-5x=\frac{299}{2}$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=\frac{299}{2}+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=\frac{299}{2}+\frac{25}{4}$$

Add \frac{299}{2} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-5x+\frac{25}{4}=\frac{623}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{623}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{623}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{623}}{2} x-\frac{5}{2}=-\frac{\sqrt{623}}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{\sqrt{623}+5}{2} x=\frac{5-\sqrt{623}}{2}$$