Solve m^2+60=-16m

$$m ^ { 2 } + 60 = - 16 m$$

Solve for m

$$m=-10$$

$$m=-6$$

Steps Using Factoring

Add 16m to both sides.

$$m^{2}+60+16m=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$m^{2}+16m+60=0$$

To solve the equation, factor m^{2}+16m+60 using formula m^{2}+\left(a+b\right)m+ab=\left(m+a\right)\left(m+b\right). To find a and b, set up a system to be solved.

$$a+b=16 ab=60$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 60.

$$1,60 2,30 3,20 4,15 5,12 6,10$$

Calculate the sum for each pair.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

The solution is the pair that gives sum 16.

$$a=6 b=10$$

Rewrite factored expression \left(m+a\right)\left(m+b\right) using the obtained values.

$$\left(m+6\right)\left(m+10\right)$$

To find equation solutions, solve m+6=0 and m+10=0.

$$m=-6 m=-10$$

Steps Using Factoring By Grouping

Add 16m to both sides.

$$m^{2}+60+16m=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$m^{2}+16m+60=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as m^{2}+am+bm+60. To find a and b, set up a system to be solved.

$$a+b=16 ab=1\times 60=60$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 60.

$$1,60 2,30 3,20 4,15 5,12 6,10$$

Calculate the sum for each pair.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

The solution is the pair that gives sum 16.

$$a=6 b=10$$

Rewrite m^{2}+16m+60 as \left(m^{2}+6m\right)+\left(10m+60\right).

$$\left(m^{2}+6m\right)+\left(10m+60\right)$$

Factor out m in the first and 10 in the second group.

$$m\left(m+6\right)+10\left(m+6\right)$$

Factor out common term m+6 by using distributive property.

$$\left(m+6\right)\left(m+10\right)$$

To find equation solutions, solve m+6=0 and m+10=0.

$$m=-6 m=-10$$

Steps Using the Quadratic Formula

Add 16m to both sides.

$$m^{2}+60+16m=0$$

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.

$$m^{2}+16m+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-16±\sqrt{16^{2}-4\times 60}}{2}$$

Square 16.

$$m=\frac{-16±\sqrt{256-4\times 60}}{2}$$

Multiply -4 times 60.

$$m=\frac{-16±\sqrt{256-240}}{2}$$

Add 256 to -240.

$$m=\frac{-16±\sqrt{16}}{2}$$

Take the square root of 16.

$$m=\frac{-16±4}{2}$$

Now solve the equation m=\frac{-16±4}{2} when ± is plus. Add -16 to 4.

$$m=-\frac{12}{2}$$

Divide -12 by 2.

$$m=-6$$

Now solve the equation m=\frac{-16±4}{2} when ± is minus. Subtract 4 from -16.

$$m=-\frac{20}{2}$$

Divide -20 by 2.

$$m=-10$$

The equation is now solved.

$$m=-6 m=-10$$

Steps for Completing the Square

Add 16m to both sides.

$$m^{2}+60+16m=0$$

Subtract 60 from both sides. Anything subtracted from zero gives its negation.

$$m^{2}+16m=-60$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+16m+8^{2}=-60+8^{2}$$

Square 8.

$$m^{2}+16m+64=-60+64$$

Add -60 to 64.

$$m^{2}+16m+64=4$$

Factor m^{2}+16m+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}.

$$\left(m+8\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+8\right)^{2}}=\sqrt{4}$$

Simplify.

$$m+8=2 m+8=-2$$

Subtract 8 from both sides of the equation.

$$m=-6 m=-10$$