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

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

Solve for m

$$m = \frac{\sqrt{21} - 1}{2} \approx 1.791287847$$

$$m=\frac{-\sqrt{21}-1}{2}\approx -2.791287847$$

Steps Using the Quadratic Formula

Subtract m^{2} from both sides.

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

Subtract 2m from both sides.

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

Combine m and -2m to get -m.

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

Add 3 to both sides.

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

Add 2 and 3 to get 5.

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

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.

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

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

$$m=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$m=\frac{-\left(-1\right)±\sqrt{1+4\times 5}}{2\left(-1\right)}$$

Multiply 4 times 5.

$$m=\frac{-\left(-1\right)±\sqrt{1+20}}{2\left(-1\right)}$$

Add 1 to 20.

$$m=\frac{-\left(-1\right)±\sqrt{21}}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$m=\frac{1±\sqrt{21}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{1±\sqrt{21}}{-2}$$

Now solve the equation m=\frac{1±\sqrt{21}}{-2} when ± is plus. Add 1 to \sqrt{21}.

$$m=\frac{\sqrt{21}+1}{-2}$$

Divide 1+\sqrt{21} by -2.

$$m=\frac{-\sqrt{21}-1}{2}$$

Now solve the equation m=\frac{1±\sqrt{21}}{-2} when ± is minus. Subtract \sqrt{21} from 1.

$$m=\frac{1-\sqrt{21}}{-2}$$

Divide 1-\sqrt{21} by -2.

$$m=\frac{\sqrt{21}-1}{2}$$

The equation is now solved.

$$m=\frac{-\sqrt{21}-1}{2} m=\frac{\sqrt{21}-1}{2}$$

Steps for Completing the Square

Subtract m^{2} from both sides.

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

Subtract 2m from both sides.

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

Combine m and -2m to get -m.

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

Subtract 2 from both sides.

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

Subtract 2 from -3 to get -5.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide -1 by -1.

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

Divide -5 by -1.

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

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+m+\frac{1}{4}=5+\frac{1}{4}$$

Add 5 to \frac{1}{4}.

$$m^{2}+m+\frac{1}{4}=\frac{21}{4}$$

Factor m^{2}+m+\frac{1}{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}.

$$\left(m+\frac{1}{2}\right)^{2}=\frac{21}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{1}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

Simplify.

$$m+\frac{1}{2}=\frac{\sqrt{21}}{2} m+\frac{1}{2}=-\frac{\sqrt{21}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$m=\frac{\sqrt{21}-1}{2} m=\frac{-\sqrt{21}-1}{2}$$