Solve 36=108+c^2-18c

$$36 = 108+ { c }^{ 2 } -18c$$

Solve for c

$$c=6$$

$$c=12$$

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

$$108+c^{2}-18c=36$$

Subtract 36 from both sides.

$$108+c^{2}-18c-36=0$$

Subtract 36 from 108 to get 72.

$$72+c^{2}-18c=0$$

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

$$c^{2}-18c+72=0$$

To solve the equation, factor c^{2}-18c+72 using formula c^{2}+\left(a+b\right)c+ab=\left(c+a\right)\left(c+b\right). To find a and b, set up a system to be solved.

$$a+b=-18 ab=72$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -18.

$$a=-12 b=-6$$

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

$$\left(c-12\right)\left(c-6\right)$$

To find equation solutions, solve c-12=0 and c-6=0.

$$c=12 c=6$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$108+c^{2}-18c=36$$

Subtract 36 from both sides.

$$108+c^{2}-18c-36=0$$

Subtract 36 from 108 to get 72.

$$72+c^{2}-18c=0$$

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

$$c^{2}-18c+72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as c^{2}+ac+bc+72. To find a and b, set up a system to be solved.

$$a+b=-18 ab=1\times 72=72$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -18.

$$a=-12 b=-6$$

Rewrite c^{2}-18c+72 as \left(c^{2}-12c\right)+\left(-6c+72\right).

$$\left(c^{2}-12c\right)+\left(-6c+72\right)$$

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

$$c\left(c-12\right)-6\left(c-12\right)$$

Factor out common term c-12 by using distributive property.

$$\left(c-12\right)\left(c-6\right)$$

To find equation solutions, solve c-12=0 and c-6=0.

$$c=12 c=6$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$108+c^{2}-18c=36$$

Subtract 36 from both sides.

$$108+c^{2}-18c-36=0$$

Subtract 36 from 108 to get 72.

$$72+c^{2}-18c=0$$

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.

$$c^{2}-18c+72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 72}}{2}$$

Square -18.

$$c=\frac{-\left(-18\right)±\sqrt{324-4\times 72}}{2}$$

Multiply -4 times 72.

$$c=\frac{-\left(-18\right)±\sqrt{324-288}}{2}$$

Add 324 to -288.

$$c=\frac{-\left(-18\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$c=\frac{-\left(-18\right)±6}{2}$$

The opposite of -18 is 18.

$$c=\frac{18±6}{2}$$

Now solve the equation c=\frac{18±6}{2} when ± is plus. Add 18 to 6.

$$c=\frac{24}{2}$$

Divide 24 by 2.

$$c=12$$

Now solve the equation c=\frac{18±6}{2} when ± is minus. Subtract 6 from 18.

$$c=\frac{12}{2}$$

Divide 12 by 2.

$$c=6$$

The equation is now solved.

$$c=12 c=6$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$108+c^{2}-18c=36$$

Subtract 108 from both sides.

$$c^{2}-18c=36-108$$

Subtract 108 from 36 to get -72.

$$c^{2}-18c=-72$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}-18c+\left(-9\right)^{2}=-72+\left(-9\right)^{2}$$

Square -9.

$$c^{2}-18c+81=-72+81$$

Add -72 to 81.

$$c^{2}-18c+81=9$$

Factor c^{2}-18c+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(c-9\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c-9\right)^{2}}=\sqrt{9}$$

Simplify.

$$c-9=3 c-9=-3$$

Add 9 to both sides of the equation.

$$c=12 c=6$$