Solve 36-(-m-2)^2=0

$$36 - ( - m - 2 ) ^ { 2 } = 0$$

Solve for m

$$m=-8$$

$$m=4$$

Steps Using Factoring By Grouping

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

$$36-\left(\left(-m\right)^{2}-4\left(-m\right)+4\right)=0$$

Calculate -m to the power of 2 and get m^{2}.

$$36-\left(m^{2}-4\left(-m\right)+4\right)=0$$

Multiply -4 and -1 to get 4.

$$36-\left(m^{2}+4m+4\right)=0$$

To find the opposite of m^{2}+4m+4, find the opposite of each term.

$$36-m^{2}-4m-4=0$$

Subtract 4 from 36 to get 32.

$$32-m^{2}-4m=0$$

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

$$-m^{2}-4m+32=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -m^{2}+am+bm+32. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-32=-32$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -32.

$$1,-32 2,-16 4,-8$$

Calculate the sum for each pair.

$$1-32=-31 2-16=-14 4-8=-4$$

The solution is the pair that gives sum -4.

$$a=4 b=-8$$

Rewrite -m^{2}-4m+32 as \left(-m^{2}+4m\right)+\left(-8m+32\right).

$$\left(-m^{2}+4m\right)+\left(-8m+32\right)$$

Factor out m in the first and 8 in the second group.

$$m\left(-m+4\right)+8\left(-m+4\right)$$

Factor out common term -m+4 by using distributive property.

$$\left(-m+4\right)\left(m+8\right)$$

To find equation solutions, solve -m+4=0 and m+8=0.

$$m=4 m=-8$$

Steps Using the Quadratic Formula

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

$$36-\left(\left(-m\right)^{2}-4\left(-m\right)+4\right)=0$$

Calculate -m to the power of 2 and get m^{2}.

$$36-\left(m^{2}-4\left(-m\right)+4\right)=0$$

Multiply -4 and -1 to get 4.

$$36-\left(m^{2}+4m+4\right)=0$$

To find the opposite of m^{2}+4m+4, find the opposite of each term.

$$36-m^{2}-4m-4=0$$

Subtract 4 from 36 to get 32.

$$32-m^{2}-4m=0$$

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.

$$-m^{2}-4m+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 32}}{2\left(-1\right)}$$

Square -4.

$$m=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 32}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$m=\frac{-\left(-4\right)±\sqrt{16+4\times 32}}{2\left(-1\right)}$$

Multiply 4 times 32.

$$m=\frac{-\left(-4\right)±\sqrt{16+128}}{2\left(-1\right)}$$

Add 16 to 128.

$$m=\frac{-\left(-4\right)±\sqrt{144}}{2\left(-1\right)}$$

Take the square root of 144.

$$m=\frac{-\left(-4\right)±12}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$m=\frac{4±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{4±12}{-2}$$

Now solve the equation m=\frac{4±12}{-2} when ± is plus. Add 4 to 12.

$$m=\frac{16}{-2}$$

Divide 16 by -2.

$$m=-8$$

Now solve the equation m=\frac{4±12}{-2} when ± is minus. Subtract 12 from 4.

$$m=-\frac{8}{-2}$$

Divide -8 by -2.

$$m=4$$

The equation is now solved.

$$m=-8 m=4$$

Steps for Completing the Square

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

$$36-\left(\left(-m\right)^{2}-4\left(-m\right)+4\right)=0$$

Calculate -m to the power of 2 and get m^{2}.

$$36-\left(m^{2}-4\left(-m\right)+4\right)=0$$

Multiply -4 and -1 to get 4.

$$36-\left(m^{2}+4m+4\right)=0$$

To find the opposite of m^{2}+4m+4, find the opposite of each term.

$$36-m^{2}-4m-4=0$$

Subtract 4 from 36 to get 32.

$$32-m^{2}-4m=0$$

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$-m^{2}-4m=-32$$

Divide both sides by -1.

$$\frac{-m^{2}-4m}{-1}=-\frac{32}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$m^{2}+\left(-\frac{4}{-1}\right)m=-\frac{32}{-1}$$

Divide -4 by -1.

$$m^{2}+4m=-\frac{32}{-1}$$

Divide -32 by -1.

$$m^{2}+4m=32$$

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

$$m^{2}+4m+2^{2}=32+2^{2}$$

Square 2.

$$m^{2}+4m+4=32+4$$

Add 32 to 4.

$$m^{2}+4m+4=36$$

Factor m^{2}+4m+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}.

$$\left(m+2\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+2\right)^{2}}=\sqrt{36}$$

Simplify.

$$m+2=6 m+2=-6$$

Subtract 2 from both sides of the equation.

$$m=4 m=-8$$