Solve 1/3x(x+80)=x^2+200

$$\frac { 1 } { 3 } x ( x + 80 ) = x ^ { 2 } + 200$$

Solve for x

$$x=10$$

$$x=30$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{3}x by x+80.

$$\frac{1}{3}x^{2}+\frac{80}{3}x=x^{2}+200$$

Subtract x^{2} from both sides.

$$\frac{1}{3}x^{2}+\frac{80}{3}x-x^{2}=200$$

Combine \frac{1}{3}x^{2} and -x^{2} to get -\frac{2}{3}x^{2}.

$$-\frac{2}{3}x^{2}+\frac{80}{3}x=200$$

Subtract 200 from both sides.

$$-\frac{2}{3}x^{2}+\frac{80}{3}x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{2}{3} for a, \frac{80}{3} for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{80}{3}±\sqrt{\left(\frac{80}{3}\right)^{2}-4\left(-\frac{2}{3}\right)\left(-200\right)}}{2\left(-\frac{2}{3}\right)}$$

Square \frac{80}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{80}{3}±\sqrt{\frac{6400}{9}-4\left(-\frac{2}{3}\right)\left(-200\right)}}{2\left(-\frac{2}{3}\right)}$$

Multiply -4 times -\frac{2}{3}.

$$x=\frac{-\frac{80}{3}±\sqrt{\frac{6400}{9}+\frac{8}{3}\left(-200\right)}}{2\left(-\frac{2}{3}\right)}$$

Multiply \frac{8}{3} times -200.

$$x=\frac{-\frac{80}{3}±\sqrt{\frac{6400}{9}-\frac{1600}{3}}}{2\left(-\frac{2}{3}\right)}$$

Add \frac{6400}{9} to -\frac{1600}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{80}{3}±\sqrt{\frac{1600}{9}}}{2\left(-\frac{2}{3}\right)}$$

Take the square root of \frac{1600}{9}.

$$x=\frac{-\frac{80}{3}±\frac{40}{3}}{2\left(-\frac{2}{3}\right)}$$

Multiply 2 times -\frac{2}{3}.

$$x=\frac{-\frac{80}{3}±\frac{40}{3}}{-\frac{4}{3}}$$

Now solve the equation x=\frac{-\frac{80}{3}±\frac{40}{3}}{-\frac{4}{3}} when ± is plus. Add -\frac{80}{3} to \frac{40}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{40}{3}}{-\frac{4}{3}}$$

Divide -\frac{40}{3} by -\frac{4}{3} by multiplying -\frac{40}{3} by the reciprocal of -\frac{4}{3}.

$$x=10$$

Now solve the equation x=\frac{-\frac{80}{3}±\frac{40}{3}}{-\frac{4}{3}} when ± is minus. Subtract \frac{40}{3} from -\frac{80}{3} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{40}{-\frac{4}{3}}$$

Divide -40 by -\frac{4}{3} by multiplying -40 by the reciprocal of -\frac{4}{3}.

$$x=30$$

The equation is now solved.

$$x=10 x=30$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{3}x by x+80.

$$\frac{1}{3}x^{2}+\frac{80}{3}x=x^{2}+200$$

Subtract x^{2} from both sides.

$$\frac{1}{3}x^{2}+\frac{80}{3}x-x^{2}=200$$

Combine \frac{1}{3}x^{2} and -x^{2} to get -\frac{2}{3}x^{2}.

$$-\frac{2}{3}x^{2}+\frac{80}{3}x=200$$

Divide both sides of the equation by -\frac{2}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{2}{3}x^{2}+\frac{80}{3}x}{-\frac{2}{3}}=\frac{200}{-\frac{2}{3}}$$

Dividing by -\frac{2}{3} undoes the multiplication by -\frac{2}{3}.

$$x^{2}+\frac{\frac{80}{3}}{-\frac{2}{3}}x=\frac{200}{-\frac{2}{3}}$$

Divide \frac{80}{3} by -\frac{2}{3} by multiplying \frac{80}{3} by the reciprocal of -\frac{2}{3}.

$$x^{2}-40x=\frac{200}{-\frac{2}{3}}$$

Divide 200 by -\frac{2}{3} by multiplying 200 by the reciprocal of -\frac{2}{3}.

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

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.

$$x=30 x=10$$