Solve (x+3)^2+13(x+3)+36=0

$$( x + 3 ) ^ { 2 } + 13 ( x + 3 ) + 36 = 0$$

Solve for x

$$x=-12$$

$$x=-7$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+13\left(x+3\right)+36=0$$

Use the distributive property to multiply 13 by x+3.

$$x^{2}+6x+9+13x+39+36=0$$

Combine 6x and 13x to get 19x.

$$x^{2}+19x+9+39+36=0$$

Add 9 and 39 to get 48.

$$x^{2}+19x+48+36=0$$

Add 48 and 36 to get 84.

$$x^{2}+19x+84=0$$

To solve the equation, factor x^{2}+19x+84 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=19 ab=84$$

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 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Calculate the sum for each pair.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

The solution is the pair that gives sum 19.

$$a=7 b=12$$

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

$$\left(x+7\right)\left(x+12\right)$$

To find equation solutions, solve x+7=0 and x+12=0.

$$x=-7 x=-12$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+13\left(x+3\right)+36=0$$

Use the distributive property to multiply 13 by x+3.

$$x^{2}+6x+9+13x+39+36=0$$

Combine 6x and 13x to get 19x.

$$x^{2}+19x+9+39+36=0$$

Add 9 and 39 to get 48.

$$x^{2}+19x+48+36=0$$

Add 48 and 36 to get 84.

$$x^{2}+19x+84=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+84. To find a and b, set up a system to be solved.

$$a+b=19 ab=1\times 84=84$$

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 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Calculate the sum for each pair.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

The solution is the pair that gives sum 19.

$$a=7 b=12$$

Rewrite x^{2}+19x+84 as \left(x^{2}+7x\right)+\left(12x+84\right).

$$\left(x^{2}+7x\right)+\left(12x+84\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(x+7\right)+12\left(x+7\right)$$

Factor out common term x+7 by using distributive property.

$$\left(x+7\right)\left(x+12\right)$$

To find equation solutions, solve x+7=0 and x+12=0.

$$x=-7 x=-12$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+13\left(x+3\right)+36=0$$

Use the distributive property to multiply 13 by x+3.

$$x^{2}+6x+9+13x+39+36=0$$

Combine 6x and 13x to get 19x.

$$x^{2}+19x+9+39+36=0$$

Add 9 and 39 to get 48.

$$x^{2}+19x+48+36=0$$

Add 48 and 36 to get 84.

$$x^{2}+19x+84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 19 for b, and 84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\times 84}}{2}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\times 84}}{2}$$

Multiply -4 times 84.

$$x=\frac{-19±\sqrt{361-336}}{2}$$

Add 361 to -336.

$$x=\frac{-19±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-19±5}{2}$$

Now solve the equation x=\frac{-19±5}{2} when ± is plus. Add -19 to 5.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

Now solve the equation x=\frac{-19±5}{2} when ± is minus. Subtract 5 from -19.

$$x=-\frac{24}{2}$$

Divide -24 by 2.

$$x=-12$$

The equation is now solved.

$$x=-7 x=-12$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+13\left(x+3\right)+36=0$$

Use the distributive property to multiply 13 by x+3.

$$x^{2}+6x+9+13x+39+36=0$$

Combine 6x and 13x to get 19x.

$$x^{2}+19x+9+39+36=0$$

Add 9 and 39 to get 48.

$$x^{2}+19x+48+36=0$$

Add 48 and 36 to get 84.

$$x^{2}+19x+84=0$$

Subtract 84 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+19x=-84$$

Divide 19, the coefficient of the x term, by 2 to get \frac{19}{2}. Then add the square of \frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+19x+\left(\frac{19}{2}\right)^{2}=-84+\left(\frac{19}{2}\right)^{2}$$

Square \frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+19x+\frac{361}{4}=-84+\frac{361}{4}$$

Add -84 to \frac{361}{4}.

$$x^{2}+19x+\frac{361}{4}=\frac{25}{4}$$

Factor x^{2}+19x+\frac{361}{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}.

$$\left(x+\frac{19}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{19}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x+\frac{19}{2}=\frac{5}{2} x+\frac{19}{2}=-\frac{5}{2}$$

Subtract \frac{19}{2} from both sides of the equation.

$$x=-7 x=-12$$