Finite Math Examples

$2 x^{6} - 5 x^{3} - 12 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$2 {(x^{3})}^{2} - 5 x^{3} - 12 = 0$

Step 1.2

Let $u = x^{3}$ . Substitute $u$ for all occurrences of $x^{3}$ .

$2 u^{2} - 5 u - 12 = 0$

Step 1.3

Factor by grouping.

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Step 1.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = - 5$ .

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Step 1.3.1.1

Factor $- 5$ out of $- 5 u$ .

$2 u^{2} - 5 u - 12 = 0$

Step 1.3.1.2

Rewrite $- 5$ as $3$ plus $- 8$

$2 u^{2} + (3 - 8) u - 12 = 0$

Step 1.3.1.3

Apply the distributive property.

$2 u^{2} + 3 u - 8 u - 12 = 0$

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

$(2 u^{2} + 3 u) - 8 u - 12 = 0$

Step 1.3.2.2

Factor out the greatest common factor (GCF) from each group.

$u (2 u + 3) - 4 (2 u + 3) = 0$

Step 1.3.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 3$ .

$(2 u + 3) (u - 4) = 0$

Step 1.4

Replace all occurrences of $u$ with $x^{3}$ .

$(2 x^{3} + 3) (x^{3} - 4) = 0$

Step 2

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

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

$x^{3} - 4 = 0$

Step 3

Set $2 x^{3} + 3$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $2 x^{3} + 3$ equal to $0$ .

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

Step 3.2

Solve $2 x^{3} + 3 = 0$ for $x$ .

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Step 3.2.1

Subtract $3$ from both sides of the equation.

$2 x^{3} = - 3$

Step 3.2.2

Divide each term in $2 x^{3} = - 3$ by $2$ and simplify.

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Step 3.2.2.1

Divide each term in $2 x^{3} = - 3$ by $2$ .

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.2.1.2

Divide $x^{3}$ by $1$ .

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

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

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

Step 3.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{- \frac{3}{2}}$

Step 3.2.4

Simplify $\sqrt[3]{- \frac{3}{2}}$ .

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Step 3.2.4.1

Rewrite $- \frac{3}{2}$ as ${({(- 1)}^{3})}^{3} \frac{3}{2}$ .

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Step 3.2.4.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \frac{3}{2}}$

Step 3.2.4.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{3}{2}}$

Step 3.2.4.2

Pull terms out from under the radical.

$x = {(- 1)}^{3} \sqrt[3]{\frac{3}{2}}$

Step 3.2.4.3

Raise $- 1$ to the power of $3$ .

$x = - \sqrt[3]{\frac{3}{2}}$

Step 3.2.4.4

Rewrite $\sqrt[3]{\frac{3}{2}}$ as $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ .

$x = - \frac{\sqrt[3]{3}}{\sqrt[3]{2}}$

Step 3.2.4.5

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = - (\frac{\sqrt[3]{3}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}})$

Step 3.2.4.6

Combine and simplify the denominator.

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Step 3.2.4.6.1

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 3.2.4.6.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 3.2.4.6.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 3.2.4.6.4

Add $1$ and $2$ .

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 3.2.4.6.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 3.2.4.6.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 3.2.4.6.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 3.2.4.6.5.3

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

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.2.4.6.5.4

Cancel the common factor of $3$ .

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Step 3.2.4.6.5.4.1

Cancel the common factor.

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.2.4.6.5.4.2

Rewrite the expression.

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 3.2.4.6.5.5

Evaluate the exponent.

$x = - \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2}$

Step 3.2.4.7

Simplify the numerator.

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Step 3.2.4.7.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$x = - \frac{\sqrt[3]{3} \sqrt[3]{2^{2}}}{2}$

Step 3.2.4.7.2

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

$x = - \frac{\sqrt[3]{3} \sqrt[3]{4}}{2}$

Step 3.2.4.8

Simplify the numerator.

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Step 3.2.4.8.1

Combine using the product rule for radicals.

$x = - \frac{\sqrt[3]{3 \cdot 4}}{2}$

Step 3.2.4.8.2

Multiply $3$ by $4$ .

$x = - \frac{\sqrt[3]{12}}{2}$

Step 4

Set $x^{3} - 4$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{3} - 4$ equal to $0$ .

$x^{3} - 4 = 0$

Step 4.2

Solve $x^{3} - 4 = 0$ for $x$ .

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Step 4.2.1

Add $4$ to both sides of the equation.

$x^{3} = 4$

Step 4.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{4}$

Step 5

The final solution is all the values that make $(2 x^{3} + 3) (x^{3} - 4) = 0$ true.

$x = - \frac{\sqrt[3]{12}}{2}, \sqrt[3]{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\sqrt[3]{12}}{2}, \sqrt[3]{4}$

Decimal Form:

$x = - 1.14471424 \dots, 1.58740105 \dots$