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

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

Solve for m

$$m=-2+\sqrt{2}i\approx -2+1.414213562i$$

$$m=-\sqrt{2}i-2\approx -2-1.414213562i$$

Steps Using the Quadratic Formula

Add \frac{1}{3}m^{2} to both sides.

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

Add \frac{1}{3}m to both sides.

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

Combine m and \frac{1}{3}m to get \frac{4}{3}m.

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

Subtract 1 from both sides.

$$3+\frac{4}{3}m+\frac{1}{3}m^{2}-1=0$$

Subtract 1 from 3 to get 2.

$$2+\frac{4}{3}m+\frac{1}{3}m^{2}=0$$

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.

$$\frac{1}{3}m^{2}+\frac{4}{3}m+2=0$$

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

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

Square \frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$m=\frac{-\frac{4}{3}±\sqrt{\frac{16}{9}-4\times \frac{1}{3}\times 2}}{2\times \frac{1}{3}}$$

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

$$m=\frac{-\frac{4}{3}±\sqrt{\frac{16}{9}-\frac{4}{3}\times 2}}{2\times \frac{1}{3}}$$

Multiply -\frac{4}{3} times 2.

$$m=\frac{-\frac{4}{3}±\sqrt{\frac{16}{9}-\frac{8}{3}}}{2\times \frac{1}{3}}$$

Add \frac{16}{9} to -\frac{8}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m=\frac{-\frac{4}{3}±\sqrt{-\frac{8}{9}}}{2\times \frac{1}{3}}$$

Take the square root of -\frac{8}{9}.

$$m=\frac{-\frac{4}{3}±\frac{2\sqrt{2}i}{3}}{2\times \frac{1}{3}}$$

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

$$m=\frac{-\frac{4}{3}±\frac{2\sqrt{2}i}{3}}{\frac{2}{3}}$$

Now solve the equation m=\frac{-\frac{4}{3}±\frac{2\sqrt{2}i}{3}}{\frac{2}{3}} when ± is plus. Add -\frac{4}{3} to \frac{2i\sqrt{2}}{3}.

$$m=\frac{-4+2\sqrt{2}i}{\frac{2}{3}\times 3}$$

Divide \frac{-4+2i\sqrt{2}}{3} by \frac{2}{3} by multiplying \frac{-4+2i\sqrt{2}}{3} by the reciprocal of \frac{2}{3}.

$$m=-2+\sqrt{2}i$$

Now solve the equation m=\frac{-\frac{4}{3}±\frac{2\sqrt{2}i}{3}}{\frac{2}{3}} when ± is minus. Subtract \frac{2i\sqrt{2}}{3} from -\frac{4}{3}.

$$m=\frac{-2\sqrt{2}i-4}{\frac{2}{3}\times 3}$$

Divide \frac{-4-2i\sqrt{2}}{3} by \frac{2}{3} by multiplying \frac{-4-2i\sqrt{2}}{3} by the reciprocal of \frac{2}{3}.

$$m=-\sqrt{2}i-2$$

The equation is now solved.

$$m=-2+\sqrt{2}i m=-\sqrt{2}i-2$$

Steps for Completing the Square

Add \frac{1}{3}m^{2} to both sides.

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

Add \frac{1}{3}m to both sides.

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

Combine m and \frac{1}{3}m to get \frac{4}{3}m.

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

Subtract 3 from both sides.

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

Subtract 3 from 1 to get -2.

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

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.

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

Multiply both sides by 3.

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

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

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

Divide \frac{4}{3} by \frac{1}{3} by multiplying \frac{4}{3} by the reciprocal of \frac{1}{3}.

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

Divide -2 by \frac{1}{3} by multiplying -2 by the reciprocal of \frac{1}{3}.

$$m^{2}+4m=-6$$

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

$$m^{2}+4m+2^{2}=-6+2^{2}$$

Square 2.

$$m^{2}+4m+4=-6+4$$

Add -6 to 4.

$$m^{2}+4m+4=-2$$

Factor m^{2}+4m+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$m+2=\sqrt{2}i m+2=-\sqrt{2}i$$

Subtract 2 from both sides of the equation.

$$m=-2+\sqrt{2}i m=-\sqrt{2}i-2$$