Solve 36left(m-2right)=left(m+6right)^2

$$36 \left( m-2 \right) = { \left(m+6 \right) }^{ 2 }$$

Solve for m

$$m=6$$

$$m=18$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 36 by m-2.

$$36m-72=\left(m+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+6\right)^{2}.

$$36m-72=m^{2}+12m+36$$

Subtract m^{2} from both sides.

$$36m-72-m^{2}=12m+36$$

Subtract 12m from both sides.

$$36m-72-m^{2}-12m=36$$

Combine 36m and -12m to get 24m.

$$24m-72-m^{2}=36$$

Subtract 36 from both sides.

$$24m-72-m^{2}-36=0$$

Subtract 36 from -72 to get -108.

$$24m-108-m^{2}=0$$

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

$$-m^{2}+24m-108=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -m^{2}+am+bm-108. To find a and b, set up a system to be solved.

$$a+b=24 ab=-\left(-108\right)=108$$

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 108.

$$1,108 2,54 3,36 4,27 6,18 9,12$$

Calculate the sum for each pair.

$$1+108=109 2+54=56 3+36=39 4+27=31 6+18=24 9+12=21$$

The solution is the pair that gives sum 24.

$$a=18 b=6$$

Rewrite -m^{2}+24m-108 as \left(-m^{2}+18m\right)+\left(6m-108\right).

$$\left(-m^{2}+18m\right)+\left(6m-108\right)$$

Factor out -m in the first and 6 in the second group.

$$-m\left(m-18\right)+6\left(m-18\right)$$

Factor out common term m-18 by using distributive property.

$$\left(m-18\right)\left(-m+6\right)$$

To find equation solutions, solve m-18=0 and -m+6=0.

$$m=18 m=6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 36 by m-2.

$$36m-72=\left(m+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+6\right)^{2}.

$$36m-72=m^{2}+12m+36$$

Subtract m^{2} from both sides.

$$36m-72-m^{2}=12m+36$$

Subtract 12m from both sides.

$$36m-72-m^{2}-12m=36$$

Combine 36m and -12m to get 24m.

$$24m-72-m^{2}=36$$

Subtract 36 from both sides.

$$24m-72-m^{2}-36=0$$

Subtract 36 from -72 to get -108.

$$24m-108-m^{2}=0$$

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.

$$-m^{2}+24m-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 24 for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-24±\sqrt{24^{2}-4\left(-1\right)\left(-108\right)}}{2\left(-1\right)}$$

Square 24.

$$m=\frac{-24±\sqrt{576-4\left(-1\right)\left(-108\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$m=\frac{-24±\sqrt{576+4\left(-108\right)}}{2\left(-1\right)}$$

Multiply 4 times -108.

$$m=\frac{-24±\sqrt{576-432}}{2\left(-1\right)}$$

Add 576 to -432.

$$m=\frac{-24±\sqrt{144}}{2\left(-1\right)}$$

Take the square root of 144.

$$m=\frac{-24±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{-24±12}{-2}$$

Now solve the equation m=\frac{-24±12}{-2} when ± is plus. Add -24 to 12.

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

Divide -12 by -2.

$$m=6$$

Now solve the equation m=\frac{-24±12}{-2} when ± is minus. Subtract 12 from -24.

$$m=-\frac{36}{-2}$$

Divide -36 by -2.

$$m=18$$

The equation is now solved.

$$m=6 m=18$$

Steps for Completing the Square

Use the distributive property to multiply 36 by m-2.

$$36m-72=\left(m+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+6\right)^{2}.

$$36m-72=m^{2}+12m+36$$

Subtract m^{2} from both sides.

$$36m-72-m^{2}=12m+36$$

Subtract 12m from both sides.

$$36m-72-m^{2}-12m=36$$

Combine 36m and -12m to get 24m.

$$24m-72-m^{2}=36$$

Add 72 to both sides.

$$24m-m^{2}=36+72$$

Add 36 and 72 to get 108.

$$24m-m^{2}=108$$

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.

$$-m^{2}+24m=108$$

Divide both sides by -1.

$$\frac{-m^{2}+24m}{-1}=\frac{108}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$m^{2}+\frac{24}{-1}m=\frac{108}{-1}$$

Divide 24 by -1.

$$m^{2}-24m=\frac{108}{-1}$$

Divide 108 by -1.

$$m^{2}-24m=-108$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-24m+\left(-12\right)^{2}=-108+\left(-12\right)^{2}$$

Square -12.

$$m^{2}-24m+144=-108+144$$

Add -108 to 144.

$$m^{2}-24m+144=36$$

Factor m^{2}-24m+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-12\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-12\right)^{2}}=\sqrt{36}$$

Simplify.

$$m-12=6 m-12=-6$$

Add 12 to both sides of the equation.

$$m=18 m=6$$