Solve 506.2=R(R+200)

$$506.2 = R ( R + 200 )$$

Solve for R

$$R=\frac{\sqrt{262655}}{5}-100\approx 2.499756097$$

$$R=-\frac{\sqrt{262655}}{5}-100\approx -202.499756097$$

Steps Using the Quadratic Formula

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

$$506.2=R^{2}+200R$$

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

$$R^{2}+200R=506.2$$

Subtract 506.2 from both sides.

$$R^{2}+200R-506.2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 200 for b, and -506.2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$R=\frac{-200±\sqrt{200^{2}-4\left(-506.2\right)}}{2}$$

Square 200.

$$R=\frac{-200±\sqrt{40000-4\left(-506.2\right)}}{2}$$

Multiply -4 times -506.2.

$$R=\frac{-200±\sqrt{40000+2024.8}}{2}$$

Add 40000 to 2024.8.

$$R=\frac{-200±\sqrt{42024.8}}{2}$$

Take the square root of 42024.8.

$$R=\frac{-200±\frac{2\sqrt{262655}}{5}}{2}$$

Now solve the equation R=\frac{-200±\frac{2\sqrt{262655}}{5}}{2} when ± is plus. Add -200 to \frac{2\sqrt{262655}}{5}.

$$R=\frac{\frac{2\sqrt{262655}}{5}-200}{2}$$

Divide -200+\frac{2\sqrt{262655}}{5} by 2.

$$R=\frac{\sqrt{262655}}{5}-100$$

Now solve the equation R=\frac{-200±\frac{2\sqrt{262655}}{5}}{2} when ± is minus. Subtract \frac{2\sqrt{262655}}{5} from -200.

$$R=\frac{-\frac{2\sqrt{262655}}{5}-200}{2}$$

Divide -200-\frac{2\sqrt{262655}}{5} by 2.

$$R=-\frac{\sqrt{262655}}{5}-100$$

The equation is now solved.

$$R=\frac{\sqrt{262655}}{5}-100 R=-\frac{\sqrt{262655}}{5}-100$$

Steps for Completing the Square

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

$$506.2=R^{2}+200R$$

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

$$R^{2}+200R=506.2$$

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.

$$R^{2}+200R+100^{2}=506.2+100^{2}$$

Square 100.

$$R^{2}+200R+10000=506.2+10000$$

Add 506.2 to 10000.

$$R^{2}+200R+10000=10506.2$$

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}.

$$\left(R+100\right)^{2}=10506.2$$

Take the square root of both sides of the equation.

$$\sqrt{\left(R+100\right)^{2}}=\sqrt{10506.2}$$

Simplify.

$$R+100=\frac{\sqrt{262655}}{5} R+100=-\frac{\sqrt{262655}}{5}$$

Subtract 100 from both sides of the equation.

$$R=\frac{\sqrt{262655}}{5}-100 R=-\frac{\sqrt{262655}}{5}-100$$