Finite Math Examples

$2 x^{\frac{2}{3}} - 11 x^{\frac{1}{3}} - 6 = 0$

Step 1

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

$2 {(x^{\frac{1}{3}})}^{2} - 11 x^{\frac{1}{3}} - 6 = 0$

Step 2

Let $u = x^{\frac{1}{3}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{3}}$ .

$2 u^{2} - 11 u - 6 = 0$

Step 3

Factor by grouping.

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Step 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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 11$ .

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Step 3.1.1

Factor $- 11$ out of $- 11 u$ .

$2 u^{2} - 11 u - 6 = 0$

Step 3.1.2

Rewrite $- 11$ as $1$ plus $- 12$

$2 u^{2} + (1 - 12) u - 6 = 0$

Step 3.1.3

Apply the distributive property.

$2 u^{2} + 1 u - 12 u - 6 = 0$

Step 3.1.4

Multiply $u$ by $1$ .

$2 u^{2} + u - 12 u - 6 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(2 u^{2} + u) - 12 u - 6 = 0$

Step 3.2.2

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

$u (2 u + 1) - 6 (2 u + 1) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$(2 u + 1) (u - 6) = 0$

Step 4

Replace all occurrences of $u$ with $x^{\frac{1}{3}}$ .

$(2 x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} - 6) = 0$

Step 5

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

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

$x^{\frac{1}{3}} - 6 = 0$

Step 6

Set $2 x^{\frac{1}{3}} + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $2 x^{\frac{1}{3}} + 1$ equal to $0$ .

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

Step 6.2

Solve $2 x^{\frac{1}{3}} + 1 = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

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

Step 6.2.2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Step 6.2.3

Simplify the exponent.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

Simplify ${(2 x^{\frac{1}{3}})}^{3}$ .

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Step 6.2.3.1.1.1

Apply the product rule to $2 x^{\frac{1}{3}}$ .

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

Step 6.2.3.1.1.2

Raise $2$ to the power of $3$ .

$8 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 6.2.3.1.1.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 6.2.3.1.1.3.1

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

$8 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 6.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 6.2.3.1.1.3.2.1

Cancel the common factor.

$8 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 6.2.3.1.1.3.2.2

Rewrite the expression.

$8 x^{1} = {(- 1)}^{3}$

Step 6.2.3.1.1.4

Simplify.

$8 x = {(- 1)}^{3}$

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

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

$8 x = - 1$

Step 6.2.4

Divide each term in $8 x = - 1$ by $8$ and simplify.

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Step 6.2.4.1

Divide each term in $8 x = - 1$ by $8$ .

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 6.2.4.2

Simplify the left side.

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Step 6.2.4.2.1

Cancel the common factor of $8$ .

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Step 6.2.4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 6.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{8}$

Step 6.2.4.3

Simplify the right side.

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Step 6.2.4.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{8}$

Step 7

Set $x^{\frac{1}{3}} - 6$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x^{\frac{1}{3}} - 6$ equal to $0$ .

$x^{\frac{1}{3}} - 6 = 0$

Step 7.2

Solve $x^{\frac{1}{3}} - 6 = 0$ for $x$ .

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Step 7.2.1

Add $6$ to both sides of the equation.

$x^{\frac{1}{3}} = 6$

Step 7.2.2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = 6^{3}$

Step 7.2.3

Simplify the exponent.

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Step 7.2.3.1

Simplify the left side.

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Step 7.2.3.1.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 7.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 7.2.3.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 6^{3}$

Step 7.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 7.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 6^{3}$

Step 7.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 6^{3}$

Step 7.2.3.1.1.2

Simplify.

$x = 6^{3}$

Step 7.2.3.2

Simplify the right side.

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Step 7.2.3.2.1

Raise $6$ to the power of $3$ .

$x = 216$

Step 8

The final solution is all the values that make $(2 x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} - 6) = 0$ true.

$x = - \frac{1}{8}, 216$