Solve for x (x+3)(x+2)=x+13

$(x + 3) (x + 2) = x + 13$

Simplify $(x + 3) (x + 2)$ .

Rewrite.

$0 + 0 + (x + 3) (x + 2) = x + 13$

Simplify by adding zeros.

$(x + 3) (x + 2) = x + 13$

Expand $(x + 3) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$x (x + 2) + 3 (x + 2) = x + 13$

Apply the distributive property.

$x \cdot x + x \cdot 2 + 3 (x + 2) = x + 13$

Apply the distributive property.

$x \cdot x + x \cdot 2 + 3 x + 3 \cdot 2 = x + 13$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 + 3 x + 3 \cdot 2 = x + 13$

Move $2$ to the left of $x$ .

$x^{2} + 2 \cdot x + 3 x + 3 \cdot 2 = x + 13$

Multiply $3$ by $2$ .

$x^{2} + 2 x + 3 x + 6 = x + 13$

Add $2 x$ and $3 x$ .

$x^{2} + 5 x + 6 = x + 13$

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$x^{2} + 5 x + 6 - x = 13$

Subtract $x$ from $5 x$ .

$x^{2} + 4 x + 6 = 13$

Move all terms to the left side of the equation and simplify.

Subtract $13$ from both sides of the equation.

$x^{2} + 4 x + 6 - 13 = 0$

Subtract $13$ from $6$ .

$x^{2} + 4 x - 7 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 4$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 7)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 7}}{2 \cdot 1}$

Multiply $- 4$ by $- 7$ .

$x = \frac{- 4 \pm \sqrt{16 + 28}}{2 \cdot 1}$

Add $16$ and $28$ .

$x = \frac{- 4 \pm \sqrt{44}}{2 \cdot 1}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{- 4 \pm \sqrt{4 (11)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{11}}{2}$

Simplify $\frac{- 4 \pm 2 \sqrt{11}}{2}$ .

$x = - 2 \pm \sqrt{11}$

The final answer is the combination of both solutions.

$x = - 2 + \sqrt{11}, - 2 - \sqrt{11}$

The result can be shown in multiple forms.

Exact Form:

$x = - 2 + \sqrt{11}, - 2 - \sqrt{11}$

Decimal Form:

$x = 1.31662479 \dots, - 5.31662479 \dots$

Solve for x (x+3)(x+2)=x+13

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