Solve r-9r^2+27r-27=0

$$r - 9 r ^ { 2 } + 27 r - 27 = 0$$

Solve for r

$$r=\frac{-\sqrt{47}i+14}{9}\approx 1.555555556-0.7617394i$$

$$r=\frac{14+\sqrt{47}i}{9}\approx 1.555555556+0.7617394i$$

Steps Using the Quadratic Formula

Combine r and 27r to get 28r.

$$28r-9r^{2}-27=0$$

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.

$$-9r^{2}+28r-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 28 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-28±\sqrt{28^{2}-4\left(-9\right)\left(-27\right)}}{2\left(-9\right)}$$

Square 28.

$$r=\frac{-28±\sqrt{784-4\left(-9\right)\left(-27\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$r=\frac{-28±\sqrt{784+36\left(-27\right)}}{2\left(-9\right)}$$

Multiply 36 times -27.

$$r=\frac{-28±\sqrt{784-972}}{2\left(-9\right)}$$

Add 784 to -972.

$$r=\frac{-28±\sqrt{-188}}{2\left(-9\right)}$$

Take the square root of -188.

$$r=\frac{-28±2\sqrt{47}i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$r=\frac{-28±2\sqrt{47}i}{-18}$$

Now solve the equation r=\frac{-28±2\sqrt{47}i}{-18} when ± is plus. Add -28 to 2i\sqrt{47}.

$$r=\frac{-28+2\sqrt{47}i}{-18}$$

Divide -28+2i\sqrt{47} by -18.

$$r=\frac{-\sqrt{47}i+14}{9}$$

Now solve the equation r=\frac{-28±2\sqrt{47}i}{-18} when ± is minus. Subtract 2i\sqrt{47} from -28.

$$r=\frac{-2\sqrt{47}i-28}{-18}$$

Divide -28-2i\sqrt{47} by -18.

$$r=\frac{14+\sqrt{47}i}{9}$$

The equation is now solved.

$$r=\frac{-\sqrt{47}i+14}{9} r=\frac{14+\sqrt{47}i}{9}$$

Steps for Completing the Square

Combine r and 27r to get 28r.

$$28r-9r^{2}-27=0$$

Add 27 to both sides. Anything plus zero gives itself.

$$28r-9r^{2}=27$$

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.

$$-9r^{2}+28r=27$$

Divide both sides by -9.

$$\frac{-9r^{2}+28r}{-9}=\frac{27}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$r^{2}+\frac{28}{-9}r=\frac{27}{-9}$$

Divide 28 by -9.

$$r^{2}-\frac{28}{9}r=\frac{27}{-9}$$

Divide 27 by -9.

$$r^{2}-\frac{28}{9}r=-3$$

Divide -\frac{28}{9}, the coefficient of the x term, by 2 to get -\frac{14}{9}. Then add the square of -\frac{14}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}-\frac{28}{9}r+\left(-\frac{14}{9}\right)^{2}=-3+\left(-\frac{14}{9}\right)^{2}$$

Square -\frac{14}{9} by squaring both the numerator and the denominator of the fraction.

$$r^{2}-\frac{28}{9}r+\frac{196}{81}=-3+\frac{196}{81}$$

Add -3 to \frac{196}{81}.

$$r^{2}-\frac{28}{9}r+\frac{196}{81}=-\frac{47}{81}$$

Factor r^{2}-\frac{28}{9}r+\frac{196}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r-\frac{14}{9}\right)^{2}=-\frac{47}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r-\frac{14}{9}\right)^{2}}=\sqrt{-\frac{47}{81}}$$

Simplify.

$$r-\frac{14}{9}=\frac{\sqrt{47}i}{9} r-\frac{14}{9}=-\frac{\sqrt{47}i}{9}$$

Add \frac{14}{9} to both sides of the equation.

$$r=\frac{14+\sqrt{47}i}{9} r=\frac{-\sqrt{47}i+14}{9}$$