Calculus Examples

$3 x (x + 2) = 2$

Step 1

Divide each term in $3 x (x + 2) = 2$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 x (x + 2) = 2$ by $3$ .

$\frac{3 x (x + 2)}{3} = \frac{2}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 x (x + 2)}{3} = \frac{2}{3}$

Step 1.2.1.2

Divide $x (x + 2)$ by $1$ .

$x (x + 2) = \frac{2}{3}$

Step 1.2.2

Apply the distributive property.

$x \cdot x + x \cdot 2 = \frac{2}{3}$

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Multiply $x$ by $x$ .

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

Step 1.2.3.2

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

$x^{2} + 2 x = \frac{2}{3}$

Step 2

Subtract $\frac{2}{3}$ from both sides of the equation.

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

Step 3

Multiply through by the least common denominator $3$ , then simplify.

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Step 3.1

Apply the distributive property.

$3 x^{2} + 3 (2 x) + 3 (- \frac{2}{3}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $2$ by $3$ .

$3 x^{2} + 6 x + 3 (- \frac{2}{3}) = 0$

Step 3.2.2

Cancel the common factor of $3$ .

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Step 3.2.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$3 x^{2} + 6 x + 3 (\frac{- 2}{3}) = 0$

Step 3.2.2.2

Cancel the common factor.

$3 x^{2} + 6 x + 3 (\frac{- 2}{3}) = 0$

Step 3.2.2.3

Rewrite the expression.

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

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 3$ , $b = 6$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (3 \cdot - 2)}}{2 \cdot 3}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 3 \cdot - 2}}{2 \cdot 3}$

Step 6.1.2

Multiply $- 4 \cdot 3 \cdot - 2$ .

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Step 6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 6 \pm \sqrt{36 - 12 \cdot - 2}}{2 \cdot 3}$

Step 6.1.2.2

Multiply $- 12$ by $- 2$ .

$x = \frac{- 6 \pm \sqrt{36 + 24}}{2 \cdot 3}$

Step 6.1.3

Add $36$ and $24$ .

$x = \frac{- 6 \pm \sqrt{60}}{2 \cdot 3}$

Step 6.1.4

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 6.1.4.1

Factor $4$ out of $60$ .

$x = \frac{- 6 \pm \sqrt{4 (15)}}{2 \cdot 3}$

Step 6.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 3}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{15}}{2 \cdot 3}$

Step 6.2

Multiply $2$ by $3$ .

$x = \frac{- 6 \pm 2 \sqrt{15}}{6}$

Step 6.3

Simplify $\frac{- 6 \pm 2 \sqrt{15}}{6}$ .

$x = \frac{- 3 \pm \sqrt{15}}{3}$

Step 7

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{15}}{3}, - \frac{3 + \sqrt{15}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{15}}{3}, - \frac{3 + \sqrt{15}}{3}$

Decimal Form:

$x = 0.29099444 \dots, - 2.29099444 \dots$