Solve by Factoring (3x+3)(x+2)=1

$(3 x + 3) (x + 2) = 1$

Step 1

Subtract $1$ from both sides of the equation.

$(3 x + 3) (x + 2) - 1 = 0$

Step 2

Simplify $(3 x + 3) (x + 2) - 1$ .

Simplify each term.

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

Apply the distributive property.

$3 x (x + 2) + 3 (x + 2) - 1 = 0$

Apply the distributive property.

$3 x \cdot x + 3 x \cdot 2 + 3 (x + 2) - 1 = 0$

Apply the distributive property.

$3 x \cdot x + 3 x \cdot 2 + 3 x + 3 \cdot 2 - 1 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$3 (x \cdot x) + 3 x \cdot 2 + 3 x + 3 \cdot 2 - 1 = 0$

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot 2 + 3 x + 3 \cdot 2 - 1 = 0$

Multiply $2$ by $3$ .

$3 x^{2} + 6 x + 3 x + 3 \cdot 2 - 1 = 0$

Multiply $3$ by $2$ .

$3 x^{2} + 6 x + 3 x + 6 - 1 = 0$

Add $6 x$ and $3 x$ .

$3 x^{2} + 9 x + 6 - 1 = 0$

Subtract $1$ from $6$ .

$3 x^{2} + 9 x + 5 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 3$ , $b = 9$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (3 \cdot 5)}}{2 \cdot 3}$

Step 5

Simplify.

Simplify the numerator.

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 9 \pm \sqrt{81 - 12 \cdot 5}}{2 \cdot 3}$

Multiply $- 12$ by $5$ .

$x = \frac{- 9 \pm \sqrt{81 - 60}}{2 \cdot 3}$

Subtract $60$ from $81$ .

$x = \frac{- 9 \pm \sqrt{21}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 9 \pm \sqrt{21}}{6}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 9 \pm \sqrt{81 - 12 \cdot 5}}{2 \cdot 3}$

Multiply $- 12$ by $5$ .

$x = \frac{- 9 \pm \sqrt{81 - 60}}{2 \cdot 3}$

Subtract $60$ from $81$ .

$x = \frac{- 9 \pm \sqrt{21}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 9 \pm \sqrt{21}}{6}$

Change the $\pm$ to $+$ .

$x = \frac{- 9 + \sqrt{21}}{6}$

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 + \sqrt{21}}{6}$

Factor $- 1$ out of $\sqrt{21}$ .

$x = \frac{- 1 \cdot 9 - 1 (- \sqrt{21})}{6}$

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{21})$ .

$x = \frac{- 1 (9 - \sqrt{21})}{6}$

Move the negative in front of the fraction.

$x = - \frac{9 - \sqrt{21}}{6}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 9 \pm \sqrt{81 - 12 \cdot 5}}{2 \cdot 3}$

Multiply $- 12$ by $5$ .

$x = \frac{- 9 \pm \sqrt{81 - 60}}{2 \cdot 3}$

Subtract $60$ from $81$ .

$x = \frac{- 9 \pm \sqrt{21}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 9 \pm \sqrt{21}}{6}$

Change the $\pm$ to $-$ .

$x = \frac{- 9 - \sqrt{21}}{6}$

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 - \sqrt{21}}{6}$

Factor $- 1$ out of $- \sqrt{21}$ .

$x = \frac{- 1 \cdot 9 - (\sqrt{21})}{6}$

Factor $- 1$ out of $- 1 (9) - (\sqrt{21})$ .

$x = \frac{- 1 (9 + \sqrt{21})}{6}$

Move the negative in front of the fraction.

$x = - \frac{9 + \sqrt{21}}{6}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{9 - \sqrt{21}}{6}, - \frac{9 + \sqrt{21}}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 - \sqrt{21}}{6}, - \frac{9 + \sqrt{21}}{6}$

Decimal Form:

$x = - 0.73623738 \dots, - 2.26376261 \dots$

Solve by Factoring (3x+3)(x+2)=1

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