Solve 132=((+6+c)c+2/2

$$132 = ( ( + 6 + c ) \frac { c + 2 } { 2 }$$

Solve for c

$$c=2\sqrt{67}-4\approx 12.370705544$$

$$c=-2\sqrt{67}-4\approx -20.370705544$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$264=\left(6+c\right)\left(c+2\right)$$

Apply the distributive property by multiplying each term of 6+c by each term of c+2.

$$264=6c+12+c^{2}+2c$$

Combine 6c and 2c to get 8c.

$$264=8c+12+c^{2}$$

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

$$8c+12+c^{2}=264$$

Subtract 264 from both sides.

$$8c+12+c^{2}-264=0$$

Subtract 264 from 12 to get -252.

$$8c-252+c^{2}=0$$

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.

$$c^{2}+8c-252=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -252 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-8±\sqrt{8^{2}-4\left(-252\right)}}{2}$$

Square 8.

$$c=\frac{-8±\sqrt{64-4\left(-252\right)}}{2}$$

Multiply -4 times -252.

$$c=\frac{-8±\sqrt{64+1008}}{2}$$

Add 64 to 1008.

$$c=\frac{-8±\sqrt{1072}}{2}$$

Take the square root of 1072.

$$c=\frac{-8±4\sqrt{67}}{2}$$

Now solve the equation c=\frac{-8±4\sqrt{67}}{2} when ± is plus. Add -8 to 4\sqrt{67}.

$$c=\frac{4\sqrt{67}-8}{2}$$

Divide -8+4\sqrt{67} by 2.

$$c=2\sqrt{67}-4$$

Now solve the equation c=\frac{-8±4\sqrt{67}}{2} when ± is minus. Subtract 4\sqrt{67} from -8.

$$c=\frac{-4\sqrt{67}-8}{2}$$

Divide -8-4\sqrt{67} by 2.

$$c=-2\sqrt{67}-4$$

The equation is now solved.

$$c=2\sqrt{67}-4 c=-2\sqrt{67}-4$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$264=\left(6+c\right)\left(c+2\right)$$

Apply the distributive property by multiplying each term of 6+c by each term of c+2.

$$264=6c+12+c^{2}+2c$$

Combine 6c and 2c to get 8c.

$$264=8c+12+c^{2}$$

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

$$8c+12+c^{2}=264$$

Subtract 12 from both sides.

$$8c+c^{2}=264-12$$

Subtract 12 from 264 to get 252.

$$8c+c^{2}=252$$

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.

$$c^{2}+8c=252$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}+8c+4^{2}=252+4^{2}$$

Square 4.

$$c^{2}+8c+16=252+16$$

Add 252 to 16.

$$c^{2}+8c+16=268$$

Factor c^{2}+8c+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}.

$$\left(c+4\right)^{2}=268$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c+4\right)^{2}}=\sqrt{268}$$

Simplify.

$$c+4=2\sqrt{67} c+4=-2\sqrt{67}$$

Subtract 4 from both sides of the equation.

$$c=2\sqrt{67}-4 c=-2\sqrt{67}-4$$