Solve 5062=R(R+200)

$$5062 = R ( R + 200 )$$

Solve for R

$$R=\sqrt{15062}-100\approx 22.727340067$$

$$R=-\sqrt{15062}-100\approx -222.727340067$$

Steps Using the Quadratic Formula

Use the distributive property to multiply R by R+200.

$$5062=R^{2}+200R$$

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

$$R^{2}+200R=5062$$

Subtract 5062 from both sides.

$$R^{2}+200R-5062=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 200 for b, and -5062 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$R=\frac{-200±\sqrt{200^{2}-4\left(-5062\right)}}{2}$$

Square 200.

$$R=\frac{-200±\sqrt{40000-4\left(-5062\right)}}{2}$$

Multiply -4 times -5062.

$$R=\frac{-200±\sqrt{40000+20248}}{2}$$

Add 40000 to 20248.

$$R=\frac{-200±\sqrt{60248}}{2}$$

Take the square root of 60248.

$$R=\frac{-200±2\sqrt{15062}}{2}$$

Now solve the equation R=\frac{-200±2\sqrt{15062}}{2} when ± is plus. Add -200 to 2\sqrt{15062}.

$$R=\frac{2\sqrt{15062}-200}{2}$$

Divide -200+2\sqrt{15062} by 2.

$$R=\sqrt{15062}-100$$

Now solve the equation R=\frac{-200±2\sqrt{15062}}{2} when ± is minus. Subtract 2\sqrt{15062} from -200.

$$R=\frac{-2\sqrt{15062}-200}{2}$$

Divide -200-2\sqrt{15062} by 2.

$$R=-\sqrt{15062}-100$$

The equation is now solved.

$$R=\sqrt{15062}-100 R=-\sqrt{15062}-100$$

Steps for Completing the Square

Use the distributive property to multiply R by R+200.

$$5062=R^{2}+200R$$

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

$$R^{2}+200R=5062$$

Divide 200, the coefficient of the x term, by 2 to get 100. Then add the square of 100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$R^{2}+200R+100^{2}=5062+100^{2}$$

Square 100.

$$R^{2}+200R+10000=5062+10000$$

Add 5062 to 10000.

$$R^{2}+200R+10000=15062$$

Factor R^{2}+200R+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(R+100\right)^{2}=15062$$

Take the square root of both sides of the equation.

$$\sqrt{\left(R+100\right)^{2}}=\sqrt{15062}$$

Simplify.

$$R+100=\sqrt{15062} R+100=-\sqrt{15062}$$

Subtract 100 from both sides of the equation.

$$R=\sqrt{15062}-100 R=-\sqrt{15062}-100$$