Solve 5=-1/60x^2+139/60x

$$5 = - \frac { 1 } { 60 } x ^ { 2 } + \frac { 139 } { 60 } x$$

Solve for x

$$x = \frac{\sqrt{18121} + 139}{2} \approx 136.807131866$$

$$x = \frac{139 - \sqrt{18121}}{2} \approx 2.192868134$$

Steps Using the Quadratic Formula

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

$$-\frac{1}{60}x^{2}+\frac{139}{60}x=5$$

Subtract 5 from both sides.

$$-\frac{1}{60}x^{2}+\frac{139}{60}x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{60} for a, \frac{139}{60} for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{139}{60}±\sqrt{\left(\frac{139}{60}\right)^{2}-4\left(-\frac{1}{60}\right)\left(-5\right)}}{2\left(-\frac{1}{60}\right)}$$

Square \frac{139}{60} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{19321}{3600}-4\left(-\frac{1}{60}\right)\left(-5\right)}}{2\left(-\frac{1}{60}\right)}$$

Multiply -4 times -\frac{1}{60}.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{19321}{3600}+\frac{1}{15}\left(-5\right)}}{2\left(-\frac{1}{60}\right)}$$

Multiply \frac{1}{15} times -5.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{19321}{3600}-\frac{1}{3}}}{2\left(-\frac{1}{60}\right)}$$

Add \frac{19321}{3600} to -\frac{1}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{18121}{3600}}}{2\left(-\frac{1}{60}\right)}$$

Take the square root of \frac{18121}{3600}.

$$x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{2\left(-\frac{1}{60}\right)}$$

Multiply 2 times -\frac{1}{60}.

$$x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{-\frac{1}{30}}$$

Now solve the equation x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{-\frac{1}{30}} when ± is plus. Add -\frac{139}{60} to \frac{\sqrt{18121}}{60}.

$$x=\frac{\sqrt{18121}-139}{-\frac{1}{30}\times 60}$$

Divide \frac{-139+\sqrt{18121}}{60} by -\frac{1}{30} by multiplying \frac{-139+\sqrt{18121}}{60} by the reciprocal of -\frac{1}{30}.

$$x=\frac{139-\sqrt{18121}}{2}$$

Now solve the equation x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{-\frac{1}{30}} when ± is minus. Subtract \frac{\sqrt{18121}}{60} from -\frac{139}{60}.

$$x=\frac{-\sqrt{18121}-139}{-\frac{1}{30}\times 60}$$

Divide \frac{-139-\sqrt{18121}}{60} by -\frac{1}{30} by multiplying \frac{-139-\sqrt{18121}}{60} by the reciprocal of -\frac{1}{30}.

$$x=\frac{\sqrt{18121}+139}{2}$$

The equation is now solved.

$$x=\frac{139-\sqrt{18121}}{2} x=\frac{\sqrt{18121}+139}{2}$$

Steps for Completing the Square

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

$$-\frac{1}{60}x^{2}+\frac{139}{60}x=5$$

Multiply both sides by -60.

$$\frac{-\frac{1}{60}x^{2}+\frac{139}{60}x}{-\frac{1}{60}}=\frac{5}{-\frac{1}{60}}$$

Dividing by -\frac{1}{60} undoes the multiplication by -\frac{1}{60}.

$$x^{2}+\frac{\frac{139}{60}}{-\frac{1}{60}}x=\frac{5}{-\frac{1}{60}}$$

Divide \frac{139}{60} by -\frac{1}{60} by multiplying \frac{139}{60} by the reciprocal of -\frac{1}{60}.

$$x^{2}-139x=\frac{5}{-\frac{1}{60}}$$

Divide 5 by -\frac{1}{60} by multiplying 5 by the reciprocal of -\frac{1}{60}.

$$x^{2}-139x=-300$$

Divide -139, the coefficient of the x term, by 2 to get -\frac{139}{2}. Then add the square of -\frac{139}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-139x+\left(-\frac{139}{2}\right)^{2}=-300+\left(-\frac{139}{2}\right)^{2}$$

Square -\frac{139}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-139x+\frac{19321}{4}=-300+\frac{19321}{4}$$

Add -300 to \frac{19321}{4}.

$$x^{2}-139x+\frac{19321}{4}=\frac{18121}{4}$$

Factor x^{2}-139x+\frac{19321}{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}.

$$\left(x-\frac{139}{2}\right)^{2}=\frac{18121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{139}{2}\right)^{2}}=\sqrt{\frac{18121}{4}}$$

Simplify.

$$x-\frac{139}{2}=\frac{\sqrt{18121}}{2} x-\frac{139}{2}=-\frac{\sqrt{18121}}{2}$$

Add \frac{139}{2} to both sides of the equation.

$$x=\frac{\sqrt{18121}+139}{2} x=\frac{139-\sqrt{18121}}{2}$$