Solve -1/3m^2+8/3m+3+1/3m-3=3

$$- \frac { 1 } { 3 } m ^ { 2 } + \frac { 8 } { 3 } m + 3 + \frac { 1 } { 3 } m - 3 = 3$$

Solve for m

$$m = \frac{3 \sqrt{5} + 9}{2} \approx 7.854101966$$

$$m = \frac{9 - 3 \sqrt{5}}{2} \approx 1.145898034$$

Steps Using the Quadratic Formula

Combine \frac{8}{3}m and \frac{1}{3}m to get 3m.

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

Subtract 3 from 3 to get 0.

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

Subtract 3 from both sides.

$$-\frac{1}{3}m^{2}+3m-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{3} for a, 3 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-3±\sqrt{3^{2}-4\left(-\frac{1}{3}\right)\left(-3\right)}}{2\left(-\frac{1}{3}\right)}$$

Square 3.

$$m=\frac{-3±\sqrt{9-4\left(-\frac{1}{3}\right)\left(-3\right)}}{2\left(-\frac{1}{3}\right)}$$

Multiply -4 times -\frac{1}{3}.

$$m=\frac{-3±\sqrt{9+\frac{4}{3}\left(-3\right)}}{2\left(-\frac{1}{3}\right)}$$

Multiply \frac{4}{3} times -3.

$$m=\frac{-3±\sqrt{9-4}}{2\left(-\frac{1}{3}\right)}$$

Add 9 to -4.

$$m=\frac{-3±\sqrt{5}}{2\left(-\frac{1}{3}\right)}$$

Multiply 2 times -\frac{1}{3}.

$$m=\frac{-3±\sqrt{5}}{-\frac{2}{3}}$$

Now solve the equation m=\frac{-3±\sqrt{5}}{-\frac{2}{3}} when ± is plus. Add -3 to \sqrt{5}.

$$m=\frac{\sqrt{5}-3}{-\frac{2}{3}}$$

Divide -3+\sqrt{5} by -\frac{2}{3} by multiplying -3+\sqrt{5} by the reciprocal of -\frac{2}{3}.

$$m=\frac{9-3\sqrt{5}}{2}$$

Now solve the equation m=\frac{-3±\sqrt{5}}{-\frac{2}{3}} when ± is minus. Subtract \sqrt{5} from -3.

$$m=\frac{-\sqrt{5}-3}{-\frac{2}{3}}$$

Divide -3-\sqrt{5} by -\frac{2}{3} by multiplying -3-\sqrt{5} by the reciprocal of -\frac{2}{3}.

$$m=\frac{3\sqrt{5}+9}{2}$$

The equation is now solved.

$$m=\frac{9-3\sqrt{5}}{2} m=\frac{3\sqrt{5}+9}{2}$$

Steps for Completing the Square

Combine \frac{8}{3}m and \frac{1}{3}m to get 3m.

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

Subtract 3 from 3 to get 0.

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

Multiply both sides by -3.

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

Dividing by -\frac{1}{3} undoes the multiplication by -\frac{1}{3}.

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

Divide 3 by -\frac{1}{3} by multiplying 3 by the reciprocal of -\frac{1}{3}.

$$m^{2}-9m=\frac{3}{-\frac{1}{3}}$$

Divide 3 by -\frac{1}{3} by multiplying 3 by the reciprocal of -\frac{1}{3}.

$$m^{2}-9m=-9$$

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

$$m^{2}-9m+\left(-\frac{9}{2}\right)^{2}=-9+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-9m+\frac{81}{4}=-9+\frac{81}{4}$$

Add -9 to \frac{81}{4}.

$$m^{2}-9m+\frac{81}{4}=\frac{45}{4}$$

Factor m^{2}-9m+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{9}{2}\right)^{2}=\frac{45}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{9}{2}\right)^{2}}=\sqrt{\frac{45}{4}}$$

Simplify.

$$m-\frac{9}{2}=\frac{3\sqrt{5}}{2} m-\frac{9}{2}=-\frac{3\sqrt{5}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$m=\frac{3\sqrt{5}+9}{2} m=\frac{9-3\sqrt{5}}{2}$$