Solve -3r^2+4=-6r

$$- 3 r ^ { 2 } + 4 = - 6 r$$

Solve for r

$$r=\frac{\sqrt{21}}{3}+1\approx 2.527525232$$

$$r=-\frac{\sqrt{21}}{3}+1\approx -0.527525232$$

Steps Using the Quadratic Formula

Add 6r to both sides.

$$-3r^{2}+4+6r=0$$

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.

$$-3r^{2}+6r+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 6 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-6±\sqrt{6^{2}-4\left(-3\right)\times 4}}{2\left(-3\right)}$$

Square 6.

$$r=\frac{-6±\sqrt{36-4\left(-3\right)\times 4}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$r=\frac{-6±\sqrt{36+12\times 4}}{2\left(-3\right)}$$

Multiply 12 times 4.

$$r=\frac{-6±\sqrt{36+48}}{2\left(-3\right)}$$

Add 36 to 48.

$$r=\frac{-6±\sqrt{84}}{2\left(-3\right)}$$

Take the square root of 84.

$$r=\frac{-6±2\sqrt{21}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$r=\frac{-6±2\sqrt{21}}{-6}$$

Now solve the equation r=\frac{-6±2\sqrt{21}}{-6} when ± is plus. Add -6 to 2\sqrt{21}.

$$r=\frac{2\sqrt{21}-6}{-6}$$

Divide -6+2\sqrt{21} by -6.

$$r=-\frac{\sqrt{21}}{3}+1$$

Now solve the equation r=\frac{-6±2\sqrt{21}}{-6} when ± is minus. Subtract 2\sqrt{21} from -6.

$$r=\frac{-2\sqrt{21}-6}{-6}$$

Divide -6-2\sqrt{21} by -6.

$$r=\frac{\sqrt{21}}{3}+1$$

The equation is now solved.

$$r=-\frac{\sqrt{21}}{3}+1 r=\frac{\sqrt{21}}{3}+1$$

Steps for Completing the Square

Add 6r to both sides.

$$-3r^{2}+4+6r=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-3r^{2}+6r=-4$$

Divide both sides by -3.

$$\frac{-3r^{2}+6r}{-3}=-\frac{4}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$r^{2}+\frac{6}{-3}r=-\frac{4}{-3}$$

Divide 6 by -3.

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

Divide -4 by -3.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}-2r+1=\frac{4}{3}+1$$

Add \frac{4}{3} to 1.

$$r^{2}-2r+1=\frac{7}{3}$$

Factor r^{2}-2r+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r-1\right)^{2}=\frac{7}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r-1\right)^{2}}=\sqrt{\frac{7}{3}}$$

Simplify.

$$r-1=\frac{\sqrt{21}}{3} r-1=-\frac{\sqrt{21}}{3}$$

Add 1 to both sides of the equation.

$$r=\frac{\sqrt{21}}{3}+1 r=-\frac{\sqrt{21}}{3}+1$$