Solve m^2+5m+6+m^2-2m-15+4=0

$$m ^ { 2 } + 5 m + 6 + m ^ { 2 } - 2 m - 15 + 4 = 0$$

Solve for m

$$m = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$m=1$$

Steps Using Factoring By Grouping

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

$$2m^{2}+5m+6-2m-15+4=0$$

Combine 5m and -2m to get 3m.

$$2m^{2}+3m+6-15+4=0$$

Subtract 15 from 6 to get -9.

$$2m^{2}+3m-9+4=0$$

Add -9 and 4 to get -5.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2m^{2}+am+bm-5. To find a and b, set up a system to be solved.

$$a+b=3 ab=2\left(-5\right)=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -10.

$$-1,10 -2,5$$

Calculate the sum for each pair.

$$-1+10=9 -2+5=3$$

The solution is the pair that gives sum 3.

$$a=-2 b=5$$

Rewrite 2m^{2}+3m-5 as \left(2m^{2}-2m\right)+\left(5m-5\right).

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

Factor out 2m in the first and 5 in the second group.

$$2m\left(m-1\right)+5\left(m-1\right)$$

Factor out common term m-1 by using distributive property.

$$\left(m-1\right)\left(2m+5\right)$$

To find equation solutions, solve m-1=0 and 2m+5=0.

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

Steps Using the Quadratic Formula

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

$$2m^{2}+5m+6-2m-15+4=0$$

Combine 5m and -2m to get 3m.

$$2m^{2}+3m+6-15+4=0$$

Subtract 15 from 6 to get -9.

$$2m^{2}+3m-9+4=0$$

Add -9 and 4 to get -5.

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

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

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

Square 3.

$$m=\frac{-3±\sqrt{9-4\times 2\left(-5\right)}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-3±\sqrt{9-8\left(-5\right)}}{2\times 2}$$

Multiply -8 times -5.

$$m=\frac{-3±\sqrt{9+40}}{2\times 2}$$

Add 9 to 40.

$$m=\frac{-3±\sqrt{49}}{2\times 2}$$

Take the square root of 49.

$$m=\frac{-3±7}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{-3±7}{4}$$

Now solve the equation m=\frac{-3±7}{4} when ± is plus. Add -3 to 7.

$$m=\frac{4}{4}$$

Divide 4 by 4.

$$m=1$$

Now solve the equation m=\frac{-3±7}{4} when ± is minus. Subtract 7 from -3.

$$m=-\frac{10}{4}$$

Reduce the fraction \frac{-10}{4} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$2m^{2}+5m+6-2m-15+4=0$$

Combine 5m and -2m to get 3m.

$$2m^{2}+3m+6-15+4=0$$

Subtract 15 from 6 to get -9.

$$2m^{2}+3m-9+4=0$$

Add -9 and 4 to get -5.

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

Add 5 to both sides. Anything plus zero gives itself.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

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

$$m^{2}+\frac{3}{2}m+\left(\frac{3}{4}\right)^{2}=\frac{5}{2}+\left(\frac{3}{4}\right)^{2}$$

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

$$m^{2}+\frac{3}{2}m+\frac{9}{16}=\frac{5}{2}+\frac{9}{16}$$

Add \frac{5}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{3}{2}m+\frac{9}{16}=\frac{49}{16}$$

Factor m^{2}+\frac{3}{2}m+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{3}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

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

Subtract \frac{3}{4} from both sides of the equation.

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