Solve 3m^2+4m-3=-1-m^2+m-30

$$3 { m }^{ 2 } +4m-3 = -1- { m }^{ 2 } +m-30$$

Solve for m

$$m=\frac{-3+\sqrt{439}i}{8}\approx -0.375+2.619040855i$$

$$m=\frac{-\sqrt{439}i-3}{8}\approx -0.375-2.619040855i$$

Steps Using the Quadratic Formula

Subtract 30 from -1 to get -31.

$$3m^{2}+4m-3=-31-m^{2}+m$$

Subtract -31 from both sides.

$$3m^{2}+4m-3-\left(-31\right)=-m^{2}+m$$

The opposite of -31 is 31.

$$3m^{2}+4m-3+31=-m^{2}+m$$

Add m^{2} to both sides.

$$3m^{2}+4m-3+31+m^{2}=m$$

Add -3 and 31 to get 28.

$$3m^{2}+4m+28+m^{2}=m$$

Combine 3m^{2} and m^{2} to get 4m^{2}.

$$4m^{2}+4m+28=m$$

Subtract m from both sides.

$$4m^{2}+4m+28-m=0$$

Combine 4m and -m to get 3m.

$$4m^{2}+3m+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 3 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-3±\sqrt{3^{2}-4\times 4\times 28}}{2\times 4}$$

Square 3.

$$m=\frac{-3±\sqrt{9-4\times 4\times 28}}{2\times 4}$$

Multiply -4 times 4.

$$m=\frac{-3±\sqrt{9-16\times 28}}{2\times 4}$$

Multiply -16 times 28.

$$m=\frac{-3±\sqrt{9-448}}{2\times 4}$$

Add 9 to -448.

$$m=\frac{-3±\sqrt{-439}}{2\times 4}$$

Take the square root of -439.

$$m=\frac{-3±\sqrt{439}i}{2\times 4}$$

Multiply 2 times 4.

$$m=\frac{-3±\sqrt{439}i}{8}$$

Now solve the equation m=\frac{-3±\sqrt{439}i}{8} when ± is plus. Add -3 to i\sqrt{439}.

$$m=\frac{-3+\sqrt{439}i}{8}$$

Now solve the equation m=\frac{-3±\sqrt{439}i}{8} when ± is minus. Subtract i\sqrt{439} from -3.

$$m=\frac{-\sqrt{439}i-3}{8}$$

The equation is now solved.

$$m=\frac{-3+\sqrt{439}i}{8} m=\frac{-\sqrt{439}i-3}{8}$$

Steps for Completing the Square

Subtract 30 from -1 to get -31.

$$3m^{2}+4m-3=-31-m^{2}+m$$

Add m^{2} to both sides.

$$3m^{2}+4m-3+m^{2}=-31+m$$

Combine 3m^{2} and m^{2} to get 4m^{2}.

$$4m^{2}+4m-3=-31+m$$

Subtract m from both sides.

$$4m^{2}+4m-3-m=-31$$

Combine 4m and -m to get 3m.

$$4m^{2}+3m-3=-31$$

Add 3 to both sides.

$$4m^{2}+3m=-31+3$$

Add -31 and 3 to get -28.

$$4m^{2}+3m=-28$$

Divide both sides by 4.

$$\frac{4m^{2}+3m}{4}=-\frac{28}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$m^{2}+\frac{3}{4}m=-\frac{28}{4}$$

Divide -28 by 4.

$$m^{2}+\frac{3}{4}m=-7$$

Divide \frac{3}{4}, the coefficient of the x term, by 2 to get \frac{3}{8}. Then add the square of \frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{3}{4}m+\left(\frac{3}{8}\right)^{2}=-7+\left(\frac{3}{8}\right)^{2}$$

Square \frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{3}{4}m+\frac{9}{64}=-7+\frac{9}{64}$$

Add -7 to \frac{9}{64}.

$$m^{2}+\frac{3}{4}m+\frac{9}{64}=-\frac{439}{64}$$

Factor m^{2}+\frac{3}{4}m+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{3}{8}\right)^{2}=-\frac{439}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{3}{8}\right)^{2}}=\sqrt{-\frac{439}{64}}$$

Simplify.

$$m+\frac{3}{8}=\frac{\sqrt{439}i}{8} m+\frac{3}{8}=-\frac{\sqrt{439}i}{8}$$

Subtract \frac{3}{8} from both sides of the equation.

$$m=\frac{-3+\sqrt{439}i}{8} m=\frac{-\sqrt{439}i-3}{8}$$