Solve 25+R=15R^2-1500

$$25 + R = 15 R ^ { 2 } - 1500$$

Solve for R

$$R = \frac{\sqrt{91501} + 1}{30} \approx 10.116377406$$

$$R=\frac{1-\sqrt{91501}}{30}\approx -10.04971074$$

Steps Using the Quadratic Formula

Subtract 15R^{2} from both sides.

$$25+R-15R^{2}=-1500$$

Add 1500 to both sides.

$$25+R-15R^{2}+1500=0$$

Add 25 and 1500 to get 1525.

$$1525+R-15R^{2}=0$$

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.

$$-15R^{2}+R+1525=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 1 for b, and 1525 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$R=\frac{-1±\sqrt{1^{2}-4\left(-15\right)\times 1525}}{2\left(-15\right)}$$

Square 1.

$$R=\frac{-1±\sqrt{1-4\left(-15\right)\times 1525}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$R=\frac{-1±\sqrt{1+60\times 1525}}{2\left(-15\right)}$$

Multiply 60 times 1525.

$$R=\frac{-1±\sqrt{1+91500}}{2\left(-15\right)}$$

Add 1 to 91500.

$$R=\frac{-1±\sqrt{91501}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$R=\frac{-1±\sqrt{91501}}{-30}$$

Now solve the equation R=\frac{-1±\sqrt{91501}}{-30} when ± is plus. Add -1 to \sqrt{91501}.

$$R=\frac{\sqrt{91501}-1}{-30}$$

Divide -1+\sqrt{91501} by -30.

$$R=\frac{1-\sqrt{91501}}{30}$$

Now solve the equation R=\frac{-1±\sqrt{91501}}{-30} when ± is minus. Subtract \sqrt{91501} from -1.

$$R=\frac{-\sqrt{91501}-1}{-30}$$

Divide -1-\sqrt{91501} by -30.

$$R=\frac{\sqrt{91501}+1}{30}$$

The equation is now solved.

$$R=\frac{1-\sqrt{91501}}{30} R=\frac{\sqrt{91501}+1}{30}$$

Steps for Completing the Square

Subtract 15R^{2} from both sides.

$$25+R-15R^{2}=-1500$$

Subtract 25 from both sides.

$$R-15R^{2}=-1500-25$$

Subtract 25 from -1500 to get -1525.

$$R-15R^{2}=-1525$$

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.

$$-15R^{2}+R=-1525$$

Divide both sides by -15.

$$\frac{-15R^{2}+R}{-15}=-\frac{1525}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$R^{2}+\frac{1}{-15}R=-\frac{1525}{-15}$$

Divide 1 by -15.

$$R^{2}-\frac{1}{15}R=-\frac{1525}{-15}$$

Reduce the fraction \frac{-1525}{-15} to lowest terms by extracting and canceling out 5.

$$R^{2}-\frac{1}{15}R=\frac{305}{3}$$

Divide -\frac{1}{15}, the coefficient of the x term, by 2 to get -\frac{1}{30}. Then add the square of -\frac{1}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$R^{2}-\frac{1}{15}R+\left(-\frac{1}{30}\right)^{2}=\frac{305}{3}+\left(-\frac{1}{30}\right)^{2}$$

Square -\frac{1}{30} by squaring both the numerator and the denominator of the fraction.

$$R^{2}-\frac{1}{15}R+\frac{1}{900}=\frac{305}{3}+\frac{1}{900}$$

Add \frac{305}{3} to \frac{1}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$R^{2}-\frac{1}{15}R+\frac{1}{900}=\frac{91501}{900}$$

Factor R^{2}-\frac{1}{15}R+\frac{1}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(R-\frac{1}{30}\right)^{2}=\frac{91501}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(R-\frac{1}{30}\right)^{2}}=\sqrt{\frac{91501}{900}}$$

Simplify.

$$R-\frac{1}{30}=\frac{\sqrt{91501}}{30} R-\frac{1}{30}=-\frac{\sqrt{91501}}{30}$$

Add \frac{1}{30} to both sides of the equation.

$$R=\frac{\sqrt{91501}+1}{30} R=\frac{1-\sqrt{91501}}{30}$$