Solve r^2=4r^2+4r+5

$$r ^ { 2 } = 4 r ^ { 2 } + 4 r + 5$$

Solve for r

$$r=\frac{-\sqrt{11}i-2}{3}\approx -0.666666667-1.105541597i$$

$$r=\frac{-2+\sqrt{11}i}{3}\approx -0.666666667+1.105541597i$$

Steps Using the Quadratic Formula

Subtract 4r^{2} from both sides.

$$r^{2}-4r^{2}=4r+5$$

Combine r^{2} and -4r^{2} to get -3r^{2}.

$$-3r^{2}=4r+5$$

Subtract 4r from both sides.

$$-3r^{2}-4r=5$$

Subtract 5 from both sides.

$$-3r^{2}-4r-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -4 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-3\right)\left(-5\right)}}{2\left(-3\right)}$$

Square -4.

$$r=\frac{-\left(-4\right)±\sqrt{16-4\left(-3\right)\left(-5\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$r=\frac{-\left(-4\right)±\sqrt{16+12\left(-5\right)}}{2\left(-3\right)}$$

Multiply 12 times -5.

$$r=\frac{-\left(-4\right)±\sqrt{16-60}}{2\left(-3\right)}$$

Add 16 to -60.

$$r=\frac{-\left(-4\right)±\sqrt{-44}}{2\left(-3\right)}$$

Take the square root of -44.

$$r=\frac{-\left(-4\right)±2\sqrt{11}i}{2\left(-3\right)}$$

The opposite of -4 is 4.

$$r=\frac{4±2\sqrt{11}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$r=\frac{4±2\sqrt{11}i}{-6}$$

Now solve the equation r=\frac{4±2\sqrt{11}i}{-6} when ± is plus. Add 4 to 2i\sqrt{11}.

$$r=\frac{4+2\sqrt{11}i}{-6}$$

Divide 4+2i\sqrt{11} by -6.

$$r=\frac{-\sqrt{11}i-2}{3}$$

Now solve the equation r=\frac{4±2\sqrt{11}i}{-6} when ± is minus. Subtract 2i\sqrt{11} from 4.

$$r=\frac{-2\sqrt{11}i+4}{-6}$$

Divide 4-2i\sqrt{11} by -6.

$$r=\frac{-2+\sqrt{11}i}{3}$$

The equation is now solved.

$$r=\frac{-\sqrt{11}i-2}{3} r=\frac{-2+\sqrt{11}i}{3}$$

Steps for Completing the Square

Subtract 4r^{2} from both sides.

$$r^{2}-4r^{2}=4r+5$$

Combine r^{2} and -4r^{2} to get -3r^{2}.

$$-3r^{2}=4r+5$$

Subtract 4r from both sides.

$$-3r^{2}-4r=5$$

Divide both sides by -3.

$$\frac{-3r^{2}-4r}{-3}=\frac{5}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$r^{2}+\left(-\frac{4}{-3}\right)r=\frac{5}{-3}$$

Divide -4 by -3.

$$r^{2}+\frac{4}{3}r=\frac{5}{-3}$$

Divide 5 by -3.

$$r^{2}+\frac{4}{3}r=-\frac{5}{3}$$

Divide \frac{4}{3}, the coefficient of the x term, by 2 to get \frac{2}{3}. Then add the square of \frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+\frac{4}{3}r+\left(\frac{2}{3}\right)^{2}=-\frac{5}{3}+\left(\frac{2}{3}\right)^{2}$$

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+\frac{4}{3}r+\frac{4}{9}=-\frac{5}{3}+\frac{4}{9}$$

Add -\frac{5}{3} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$r^{2}+\frac{4}{3}r+\frac{4}{9}=-\frac{11}{9}$$

Factor r^{2}+\frac{4}{3}r+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+\frac{2}{3}\right)^{2}=-\frac{11}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{2}{3}\right)^{2}}=\sqrt{-\frac{11}{9}}$$

Simplify.

$$r+\frac{2}{3}=\frac{\sqrt{11}i}{3} r+\frac{2}{3}=-\frac{\sqrt{11}i}{3}$$

Subtract \frac{2}{3} from both sides of the equation.

$$r=\frac{-2+\sqrt{11}i}{3} r=\frac{-\sqrt{11}i-2}{3}$$