Solve 60-139r=12r^2

$$60 - 139 r = 12 r ^ { 2 }$$

Solve for r

$$r=\frac{5}{12}\approx 0.416666667$$

$$r=-12$$

Steps Using Factoring By Grouping

Subtract 12r^{2} from both sides.

$$60-139r-12r^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-12r^{2}-139r+60=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -12r^{2}+ar+br+60. To find a and b, set up a system to be solved.

$$a+b=-139 ab=-12\times 60=-720$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -720.

$$1,-720 2,-360 3,-240 4,-180 5,-144 6,-120 8,-90 9,-80 10,-72 12,-60 15,-48 16,-45 18,-40 20,-36 24,-30$$

Calculate the sum for each pair.

$$1-720=-719 2-360=-358 3-240=-237 4-180=-176 5-144=-139 6-120=-114 8-90=-82 9-80=-71 10-72=-62 12-60=-48 15-48=-33 16-45=-29 18-40=-22 20-36=-16 24-30=-6$$

The solution is the pair that gives sum -139.

$$a=5 b=-144$$

Rewrite -12r^{2}-139r+60 as \left(-12r^{2}+5r\right)+\left(-144r+60\right).

$$\left(-12r^{2}+5r\right)+\left(-144r+60\right)$$

Factor out -r in the first and -12 in the second group.

$$-r\left(12r-5\right)-12\left(12r-5\right)$$

Factor out common term 12r-5 by using distributive property.

$$\left(12r-5\right)\left(-r-12\right)$$

To find equation solutions, solve 12r-5=0 and -r-12=0.

$$r=\frac{5}{12} r=-12$$

Steps Using the Quadratic Formula

Subtract 12r^{2} from both sides.

$$60-139r-12r^{2}=0$$

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.

$$-12r^{2}-139r+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, -139 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-139\right)±\sqrt{\left(-139\right)^{2}-4\left(-12\right)\times 60}}{2\left(-12\right)}$$

Square -139.

$$r=\frac{-\left(-139\right)±\sqrt{19321-4\left(-12\right)\times 60}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$r=\frac{-\left(-139\right)±\sqrt{19321+48\times 60}}{2\left(-12\right)}$$

Multiply 48 times 60.

$$r=\frac{-\left(-139\right)±\sqrt{19321+2880}}{2\left(-12\right)}$$

Add 19321 to 2880.

$$r=\frac{-\left(-139\right)±\sqrt{22201}}{2\left(-12\right)}$$

Take the square root of 22201.

$$r=\frac{-\left(-139\right)±149}{2\left(-12\right)}$$

The opposite of -139 is 139.

$$r=\frac{139±149}{2\left(-12\right)}$$

Multiply 2 times -12.

$$r=\frac{139±149}{-24}$$

Now solve the equation r=\frac{139±149}{-24} when ± is plus. Add 139 to 149.

$$r=\frac{288}{-24}$$

Divide 288 by -24.

$$r=-12$$

Now solve the equation r=\frac{139±149}{-24} when ± is minus. Subtract 149 from 139.

$$r=-\frac{10}{-24}$$

Reduce the fraction \frac{-10}{-24} to lowest terms by extracting and canceling out 2.

$$r=\frac{5}{12}$$

The equation is now solved.

$$r=-12 r=\frac{5}{12}$$

Steps for Completing the Square

Subtract 12r^{2} from both sides.

$$60-139r-12r^{2}=0$$

Subtract 60 from both sides. Anything subtracted from zero gives its negation.

$$-139r-12r^{2}=-60$$

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.

$$-12r^{2}-139r=-60$$

Divide both sides by -12.

$$\frac{-12r^{2}-139r}{-12}=-\frac{60}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$r^{2}+\left(-\frac{139}{-12}\right)r=-\frac{60}{-12}$$

Divide -139 by -12.

$$r^{2}+\frac{139}{12}r=-\frac{60}{-12}$$

Divide -60 by -12.

$$r^{2}+\frac{139}{12}r=5$$

Divide \frac{139}{12}, the coefficient of the x term, by 2 to get \frac{139}{24}. Then add the square of \frac{139}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+\frac{139}{12}r+\left(\frac{139}{24}\right)^{2}=5+\left(\frac{139}{24}\right)^{2}$$

Square \frac{139}{24} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+\frac{139}{12}r+\frac{19321}{576}=5+\frac{19321}{576}$$

Add 5 to \frac{19321}{576}.

$$r^{2}+\frac{139}{12}r+\frac{19321}{576}=\frac{22201}{576}$$

Factor r^{2}+\frac{139}{12}r+\frac{19321}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+\frac{139}{24}\right)^{2}=\frac{22201}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{139}{24}\right)^{2}}=\sqrt{\frac{22201}{576}}$$

Simplify.

$$r+\frac{139}{24}=\frac{149}{24} r+\frac{139}{24}=-\frac{149}{24}$$

Subtract \frac{139}{24} from both sides of the equation.

$$r=\frac{5}{12} r=-12$$