Trigonometry Examples

$x^{3} = \frac{1}{3} \cdot (x (19 x + 14))$

Step 1

Subtract $\frac{1}{3} \cdot (x (19 x + 14))$ from both sides of the equation.

$x^{3} - \frac{1}{3} \cdot (x (19 x + 14)) = 0$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$x^{3} - \frac{1}{3} \cdot (x (19 x) + x \cdot 14) = 0$

Step 2.2

Rewrite using the commutative property of multiplication.

$x^{3} - \frac{1}{3} \cdot (19 x \cdot x + x \cdot 14) = 0$

Step 2.3

Move $14$ to the left of $x$ .

$x^{3} - \frac{1}{3} \cdot (19 x \cdot x + 14 \cdot x) = 0$

Step 2.4

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

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Step 2.4.1

Move $x$ .

$x^{3} - \frac{1}{3} \cdot (19 (x \cdot x) + 14 \cdot x) = 0$

Step 2.4.2

Multiply $x$ by $x$ .

$x^{3} - \frac{1}{3} \cdot (19 x^{2} + 14 \cdot x) = 0$

$x^{3} - \frac{1}{3} \cdot (19 x^{2} + 14 x) = 0$

Step 2.5

Apply the distributive property.

$x^{3} - \frac{1}{3} (19 x^{2}) - \frac{1}{3} (14 x) = 0$

Step 2.6

Multiply $- \frac{1}{3} (19 x^{2})$ .

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Step 2.6.1

Multiply $19$ by $- 1$ .

$x^{3} - 19 (\frac{1}{3}) x^{2} - \frac{1}{3} (14 x) = 0$

Step 2.6.2

Combine $- 19$ and $\frac{1}{3}$ .

$x^{3} + \frac{- 19}{3} x^{2} - \frac{1}{3} (14 x) = 0$

Step 2.6.3

Combine $\frac{- 19}{3}$ and $x^{2}$ .

$x^{3} + \frac{- 19 x^{2}}{3} - \frac{1}{3} (14 x) = 0$

Step 2.7

Multiply $- \frac{1}{3} (14 x)$ .

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Step 2.7.1

Multiply $14$ by $- 1$ .

$x^{3} + \frac{- 19 x^{2}}{3} - 14 (\frac{1}{3}) x = 0$

Step 2.7.2

Combine $- 14$ and $\frac{1}{3}$ .

$x^{3} + \frac{- 19 x^{2}}{3} + \frac{- 14}{3} x = 0$

Step 2.7.3

Combine $\frac{- 14}{3}$ and $x$ .

$x^{3} + \frac{- 19 x^{2}}{3} + \frac{- 14 x}{3} = 0$

Step 2.8

Simplify each term.

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Step 2.8.1

Move the negative in front of the fraction.

$x^{3} - \frac{19 x^{2}}{3} + \frac{- 14 x}{3} = 0$

Step 2.8.2

Move the negative in front of the fraction.

$x^{3} - \frac{19 x^{2}}{3} - \frac{14 x}{3} = 0$

Step 3

Factor $x$ out of $x^{3} - \frac{19 x^{2}}{3} - \frac{14 x}{3}$ .

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Step 3.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - \frac{19 x^{2}}{3} - \frac{14 x}{3} = 0$

Step 3.2

Factor $x$ out of $- \frac{19 x^{2}}{3}$ .

$x \cdot x^{2} + x (- \frac{19 x}{3}) - \frac{14 x}{3} = 0$

Step 3.3

Factor $x$ out of $- \frac{14 x}{3}$ .

$x \cdot x^{2} + x (- \frac{19 x}{3}) + x (- \frac{14}{3}) = 0$

Step 3.4

Factor $x$ out of $x \cdot x^{2} + x (- \frac{19 x}{3})$ .

$x (x^{2} - \frac{19 x}{3}) + x (- \frac{14}{3}) = 0$

Step 3.5

Factor $x$ out of $x (x^{2} - \frac{19 x}{3}) + x (- \frac{14}{3})$ .

$x (x^{2} - \frac{19 x}{3} - \frac{14}{3}) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} - \frac{19 x}{3} - \frac{14}{3} = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $x^{2} - \frac{19 x}{3} - \frac{14}{3}$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{2} - \frac{19 x}{3} - \frac{14}{3}$ equal to $0$ .

$x^{2} - \frac{19 x}{3} - \frac{14}{3} = 0$

Step 6.2

Solve $x^{2} - \frac{19 x}{3} - \frac{14}{3} = 0$ for $x$ .

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Step 6.2.1

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

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Step 6.2.1.1

Apply the distributive property.

$3 x^{2} + 3 (- \frac{19 x}{3}) + 3 (- \frac{14}{3}) = 0$

Step 6.2.1.2

Simplify.

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Step 6.2.1.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.2.1.1

Move the leading negative in $- \frac{19 x}{3}$ into the numerator.

$3 x^{2} + 3 (\frac{- 19 x}{3}) + 3 (- \frac{14}{3}) = 0$

Step 6.2.1.2.1.2

Cancel the common factor.

$3 x^{2} + 3 (\frac{- 19 x}{3}) + 3 (- \frac{14}{3}) = 0$

Step 6.2.1.2.1.3

Rewrite the expression.

$3 x^{2} - 19 x + 3 (- \frac{14}{3}) = 0$

Step 6.2.1.2.2

Cancel the common factor of $3$ .

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Step 6.2.1.2.2.1

Move the leading negative in $- \frac{14}{3}$ into the numerator.

$3 x^{2} - 19 x + 3 (\frac{- 14}{3}) = 0$

Step 6.2.1.2.2.2

Cancel the common factor.

$3 x^{2} - 19 x + 3 (\frac{- 14}{3}) = 0$

Step 6.2.1.2.2.3

Rewrite the expression.

$3 x^{2} - 19 x - 14 = 0$

Step 6.2.2

Use the quadratic formula to find the solutions.

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

Step 6.2.3

Substitute the values $a = 3$ , $b = - 19$ , and $c = - 14$ into the quadratic formula and solve for $x$ .

$\frac{19 \pm \sqrt{{(- 19)}^{2} - 4 \cdot (3 \cdot - 14)}}{2 \cdot 3}$

Step 6.2.4

Simplify.

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Step 6.2.4.1

Simplify the numerator.

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Step 6.2.4.1.1

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 3 \cdot - 14}}{2 \cdot 3}$

Step 6.2.4.1.2

Multiply $- 4 \cdot 3 \cdot - 14$ .

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Step 6.2.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{19 \pm \sqrt{361 - 12 \cdot - 14}}{2 \cdot 3}$

Step 6.2.4.1.2.2

Multiply $- 12$ by $- 14$ .

$x = \frac{19 \pm \sqrt{361 + 168}}{2 \cdot 3}$

Step 6.2.4.1.3

Add $361$ and $168$ .

$x = \frac{19 \pm \sqrt{529}}{2 \cdot 3}$

Step 6.2.4.1.4

Rewrite $529$ as $23^{2}$ .

$x = \frac{19 \pm \sqrt{23^{2}}}{2 \cdot 3}$

Step 6.2.4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{19 \pm 23}{2 \cdot 3}$

Step 6.2.4.2

Multiply $2$ by $3$ .

$x = \frac{19 \pm 23}{6}$

Step 6.2.5

The final answer is the combination of both solutions.

$x = 7, - \frac{2}{3}$

Step 7

The final solution is all the values that make $x (x^{2} - \frac{19 x}{3} - \frac{14}{3}) = 0$ true.

$x = 0, 7, - \frac{2}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, 7, - \frac{2}{3}$

Decimal Form:

$x = 0, 7, - 0.\overline{6}$