Solve 9R^2-5R/-12R^2=1-2R/-2R^2

$$\frac { 9 R ^ { 2 } - 5 R } { - 12 R ^ { 2 } } = \frac { 1 - 2 R } { - 2 R ^ { 2 } }$$

Solve for R

$$R=\frac{\sqrt{265}-7}{18}\approx 0.515490033$$

$$R=\frac{-\sqrt{265}-7}{18}\approx -1.293267811$$

Steps Using the Quadratic Formula

Variable R cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12R^{2}, the least common multiple of -12R^{2},-2R^{2}.

$$-\left(9R^{2}-5R\right)=-6\left(1-2R\right)$$

To find the opposite of 9R^{2}-5R, find the opposite of each term.

$$-9R^{2}+5R=-6\left(1-2R\right)$$

Use the distributive property to multiply -6 by 1-2R.

$$-9R^{2}+5R=-6+12R$$

Subtract -6 from both sides.

$$-9R^{2}+5R-\left(-6\right)=12R$$

The opposite of -6 is 6.

$$-9R^{2}+5R+6=12R$$

Subtract 12R from both sides.

$$-9R^{2}+5R+6-12R=0$$

Combine 5R and -12R to get -7R.

$$-9R^{2}-7R+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -7 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$R=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-9\right)\times 6}}{2\left(-9\right)}$$

Square -7.

$$R=\frac{-\left(-7\right)±\sqrt{49-4\left(-9\right)\times 6}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$R=\frac{-\left(-7\right)±\sqrt{49+36\times 6}}{2\left(-9\right)}$$

Multiply 36 times 6.

$$R=\frac{-\left(-7\right)±\sqrt{49+216}}{2\left(-9\right)}$$

Add 49 to 216.

$$R=\frac{-\left(-7\right)±\sqrt{265}}{2\left(-9\right)}$$

The opposite of -7 is 7.

$$R=\frac{7±\sqrt{265}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$R=\frac{7±\sqrt{265}}{-18}$$

Now solve the equation R=\frac{7±\sqrt{265}}{-18} when ± is plus. Add 7 to \sqrt{265}.

$$R=\frac{\sqrt{265}+7}{-18}$$

Divide 7+\sqrt{265} by -18.

$$R=\frac{-\sqrt{265}-7}{18}$$

Now solve the equation R=\frac{7±\sqrt{265}}{-18} when ± is minus. Subtract \sqrt{265} from 7.

$$R=\frac{7-\sqrt{265}}{-18}$$

Divide 7-\sqrt{265} by -18.

$$R=\frac{\sqrt{265}-7}{18}$$

The equation is now solved.

$$R=\frac{-\sqrt{265}-7}{18} R=\frac{\sqrt{265}-7}{18}$$

Steps for Completing the Square

Variable R cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12R^{2}, the least common multiple of -12R^{2},-2R^{2}.

$$-\left(9R^{2}-5R\right)=-6\left(1-2R\right)$$

To find the opposite of 9R^{2}-5R, find the opposite of each term.

$$-9R^{2}+5R=-6\left(1-2R\right)$$

Use the distributive property to multiply -6 by 1-2R.

$$-9R^{2}+5R=-6+12R$$

Subtract 12R from both sides.

$$-9R^{2}+5R-12R=-6$$

Combine 5R and -12R to get -7R.

$$-9R^{2}-7R=-6$$

Divide both sides by -9.

$$\frac{-9R^{2}-7R}{-9}=-\frac{6}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$R^{2}+\left(-\frac{7}{-9}\right)R=-\frac{6}{-9}$$

Divide -7 by -9.

$$R^{2}+\frac{7}{9}R=-\frac{6}{-9}$$

Reduce the fraction \frac{-6}{-9} to lowest terms by extracting and canceling out 3.

$$R^{2}+\frac{7}{9}R=\frac{2}{3}$$

Divide \frac{7}{9}, the coefficient of the x term, by 2 to get \frac{7}{18}. Then add the square of \frac{7}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$R^{2}+\frac{7}{9}R+\left(\frac{7}{18}\right)^{2}=\frac{2}{3}+\left(\frac{7}{18}\right)^{2}$$

Square \frac{7}{18} by squaring both the numerator and the denominator of the fraction.

$$R^{2}+\frac{7}{9}R+\frac{49}{324}=\frac{2}{3}+\frac{49}{324}$$

Add \frac{2}{3} to \frac{49}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$R^{2}+\frac{7}{9}R+\frac{49}{324}=\frac{265}{324}$$

Factor R^{2}+\frac{7}{9}R+\frac{49}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(R+\frac{7}{18}\right)^{2}=\frac{265}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(R+\frac{7}{18}\right)^{2}}=\sqrt{\frac{265}{324}}$$

Simplify.

$$R+\frac{7}{18}=\frac{\sqrt{265}}{18} R+\frac{7}{18}=-\frac{\sqrt{265}}{18}$$

Subtract \frac{7}{18} from both sides of the equation.

$$R=\frac{\sqrt{265}-7}{18} R=\frac{-\sqrt{265}-7}{18}$$