Solve pi(r+5)^2=2pir

$$\pi ( r + 5 ) ^ { 2 } = 2 \pi r$$

Solve for r

$$r=-4+3i$$

$$r=-4-3i$$

Steps Using the Quadratic Formula

Cancel out \pi on both sides.

$$\left(r+5\right)^{2}=2r$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(r+5\right)^{2}.

$$r^{2}+10r+25=2r$$

Subtract 2r from both sides.

$$r^{2}+10r+25-2r=0$$

Combine 10r and -2r to get 8r.

$$r^{2}+8r+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-8±\sqrt{8^{2}-4\times 25}}{2}$$

Square 8.

$$r=\frac{-8±\sqrt{64-4\times 25}}{2}$$

Multiply -4 times 25.

$$r=\frac{-8±\sqrt{64-100}}{2}$$

Add 64 to -100.

$$r=\frac{-8±\sqrt{-36}}{2}$$

Take the square root of -36.

$$r=\frac{-8±6i}{2}$$

Now solve the equation r=\frac{-8±6i}{2} when ± is plus. Add -8 to 6i.

$$r=\frac{-8+6i}{2}$$

Divide -8+6i by 2.

$$r=-4+3i$$

Now solve the equation r=\frac{-8±6i}{2} when ± is minus. Subtract 6i from -8.

$$r=\frac{-8-6i}{2}$$

Divide -8-6i by 2.

$$r=-4-3i$$

The equation is now solved.

$$r=-4+3i r=-4-3i$$

Steps for Completing the Square

Cancel out \pi on both sides.

$$\left(r+5\right)^{2}=2r$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(r+5\right)^{2}.

$$r^{2}+10r+25=2r$$

Subtract 2r from both sides.

$$r^{2}+10r+25-2r=0$$

Combine 10r and -2r to get 8r.

$$r^{2}+8r+25=0$$

Subtract 25 from both sides. Anything subtracted from zero gives its negation.

$$r^{2}+8r=-25$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+8r+4^{2}=-25+4^{2}$$

Square 4.

$$r^{2}+8r+16=-25+16$$

Add -25 to 16.

$$r^{2}+8r+16=-9$$

Factor r^{2}+8r+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+4\right)^{2}=-9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+4\right)^{2}}=\sqrt{-9}$$

Simplify.

$$r+4=3i r+4=-3i$$

Subtract 4 from both sides of the equation.

$$r=-4+3i r=-4-3i$$