Solve (-2m-1)^2-(m^2-2)+m=2

$$( - 2 m - 1 ) ^ { 2 } - ( m ^ { 2 } - 2 ) + m = 2$$

Solve for m

$$m=\frac{\sqrt{13}-5}{6}\approx -0.232408121$$

$$m=\frac{-\sqrt{13}-5}{6}\approx -1.434258546$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-2m-1\right)^{2}.

$$4m^{2}+4m+1-\left(m^{2}-2\right)+m=2$$

To find the opposite of m^{2}-2, find the opposite of each term.

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

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

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

Add 1 and 2 to get 3.

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

Combine 4m and m to get 5m.

$$3m^{2}+5m+3=2$$

Subtract 2 from both sides.

$$3m^{2}+5m+3-2=0$$

Subtract 2 from 3 to get 1.

$$3m^{2}+5m+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 5 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-5±\sqrt{5^{2}-4\times 3}}{2\times 3}$$

Square 5.

$$m=\frac{-5±\sqrt{25-4\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$m=\frac{-5±\sqrt{25-12}}{2\times 3}$$

Add 25 to -12.

$$m=\frac{-5±\sqrt{13}}{2\times 3}$$

Multiply 2 times 3.

$$m=\frac{-5±\sqrt{13}}{6}$$

Now solve the equation m=\frac{-5±\sqrt{13}}{6} when ± is plus. Add -5 to \sqrt{13}.

$$m=\frac{\sqrt{13}-5}{6}$$

Now solve the equation m=\frac{-5±\sqrt{13}}{6} when ± is minus. Subtract \sqrt{13} from -5.

$$m=\frac{-\sqrt{13}-5}{6}$$

The equation is now solved.

$$m=\frac{\sqrt{13}-5}{6} m=\frac{-\sqrt{13}-5}{6}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-2m-1\right)^{2}.

$$4m^{2}+4m+1-\left(m^{2}-2\right)+m=2$$

To find the opposite of m^{2}-2, find the opposite of each term.

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

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

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

Add 1 and 2 to get 3.

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

Combine 4m and m to get 5m.

$$3m^{2}+5m+3=2$$

Subtract 3 from both sides.

$$3m^{2}+5m=2-3$$

Subtract 3 from 2 to get -1.

$$3m^{2}+5m=-1$$

Divide both sides by 3.

$$\frac{3m^{2}+5m}{3}=-\frac{1}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$m^{2}+\frac{5}{3}m=-\frac{1}{3}$$

Divide \frac{5}{3}, the coefficient of the x term, by 2 to get \frac{5}{6}. Then add the square of \frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{5}{3}m+\left(\frac{5}{6}\right)^{2}=-\frac{1}{3}+\left(\frac{5}{6}\right)^{2}$$

Square \frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{5}{3}m+\frac{25}{36}=-\frac{1}{3}+\frac{25}{36}$$

Add -\frac{1}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{5}{3}m+\frac{25}{36}=\frac{13}{36}$$

Factor m^{2}+\frac{5}{3}m+\frac{25}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{5}{6}\right)^{2}=\frac{13}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{5}{6}\right)^{2}}=\sqrt{\frac{13}{36}}$$

Simplify.

$$m+\frac{5}{6}=\frac{\sqrt{13}}{6} m+\frac{5}{6}=-\frac{\sqrt{13}}{6}$$

Subtract \frac{5}{6} from both sides of the equation.

$$m=\frac{\sqrt{13}-5}{6} m=\frac{-\sqrt{13}-5}{6}$$