Solve 2(x+6)^2-5=45

$$2 ( x + 6 ) ^ { 2 } - 5 = 45$$

Solve for x

$$x=-1$$

$$x=-11$$

Steps Using Factoring By Grouping

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

$$2\left(x^{2}+12x+36\right)-5=45$$

Use the distributive property to multiply 2 by x^{2}+12x+36.

$$2x^{2}+24x+72-5=45$$

Subtract 5 from 72 to get 67.

$$2x^{2}+24x+67=45$$

Subtract 45 from both sides.

$$2x^{2}+24x+67-45=0$$

Subtract 45 from 67 to get 22.

$$2x^{2}+24x+22=0$$

Divide both sides by 2.

$$x^{2}+12x+11=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+11. To find a and b, set up a system to be solved.

$$a+b=12 ab=1\times 11=11$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=11$$

Rewrite x^{2}+12x+11 as \left(x^{2}+x\right)+\left(11x+11\right).

$$\left(x^{2}+x\right)+\left(11x+11\right)$$

Factor out x in the first and 11 in the second group.

$$x\left(x+1\right)+11\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(x+11\right)$$

To find equation solutions, solve x+1=0 and x+11=0.

$$x=-1 x=-11$$

Steps Using the Quadratic Formula

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

$$2\left(x^{2}+12x+36\right)-5=45$$

Use the distributive property to multiply 2 by x^{2}+12x+36.

$$2x^{2}+24x+72-5=45$$

Subtract 5 from 72 to get 67.

$$2x^{2}+24x+67=45$$

Subtract 45 from both sides.

$$2x^{2}+24x+67-45=0$$

Subtract 45 from 67 to get 22.

$$2x^{2}+24x+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 24 for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 2\times 22}}{2\times 2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 2\times 22}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-24±\sqrt{576-8\times 22}}{2\times 2}$$

Multiply -8 times 22.

$$x=\frac{-24±\sqrt{576-176}}{2\times 2}$$

Add 576 to -176.

$$x=\frac{-24±\sqrt{400}}{2\times 2}$$

Take the square root of 400.

$$x=\frac{-24±20}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-24±20}{4}$$

Now solve the equation x=\frac{-24±20}{4} when ± is plus. Add -24 to 20.

$$x=-\frac{4}{4}$$

Divide -4 by 4.

$$x=-1$$

Now solve the equation x=\frac{-24±20}{4} when ± is minus. Subtract 20 from -24.

$$x=-\frac{44}{4}$$

Divide -44 by 4.

$$x=-11$$

The equation is now solved.

$$x=-1 x=-11$$

Steps for Completing the Square

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

$$2\left(x^{2}+12x+36\right)-5=45$$

Use the distributive property to multiply 2 by x^{2}+12x+36.

$$2x^{2}+24x+72-5=45$$

Subtract 5 from 72 to get 67.

$$2x^{2}+24x+67=45$$

Subtract 67 from both sides.

$$2x^{2}+24x=45-67$$

Subtract 67 from 45 to get -22.

$$2x^{2}+24x=-22$$

Divide both sides by 2.

$$\frac{2x^{2}+24x}{2}=-\frac{22}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{24}{2}x=-\frac{22}{2}$$

Divide 24 by 2.

$$x^{2}+12x=-\frac{22}{2}$$

Divide -22 by 2.

$$x^{2}+12x=-11$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=-11+6^{2}$$

Square 6.

$$x^{2}+12x+36=-11+36$$

Add -11 to 36.

$$x^{2}+12x+36=25$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+6=5 x+6=-5$$

Subtract 6 from both sides of the equation.

$$x=-1 x=-11$$