Algebra Examples

$2 x^{\frac{4}{3}} - x^{\frac{2}{3}} - 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $x^{\frac{4}{3}}$ as ${(x^{\frac{2}{3}})}^{2}$ .

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

Step 1.2

Let $u = x^{\frac{2}{3}}$ . Substitute $u$ for all occurrences of $x^{\frac{2}{3}}$ .

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

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 1$ .

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Step 1.3.1.1

Factor $- 1$ out of $- u$ .

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

Step 1.3.1.2

Rewrite $- 1$ as $3$ plus $- 4$

$2 u^{2} + (3 - 4) u - 6 = 0$

Step 1.3.1.3

Apply the distributive property.

$2 u^{2} + 3 u - 4 u - 6 = 0$

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.

$(2 u^{2} + 3 u) - 4 u - 6 = 0$

Step 1.3.2.2

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

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

Step 1.3.3

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

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

Step 1.4

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

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

Step 2

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

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Subtract $3$ from both sides of the equation.

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

Step 3.2.2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

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

Step 3.2.3

Simplify the left side.

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Step 3.2.3.1

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

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Step 3.2.3.1.1

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

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

Step 3.2.3.1.2

Multiply the exponents in ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 3.2.3.1.2.1

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

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

Step 3.2.3.1.2.2

Cancel the common factor of $2$ .

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Step 3.2.3.1.2.2.1

Cancel the common factor.

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

Step 3.2.3.1.2.2.2

Rewrite the expression.

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

Step 3.2.3.1.2.3

Cancel the common factor of $3$ .

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Step 3.2.3.1.2.3.1

Cancel the common factor.

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

Step 3.2.3.1.2.3.2

Rewrite the expression.

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

Step 3.2.3.1.3

Simplify.

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

Step 3.2.3.1.4

Reorder factors in $2^{\frac{3}{2}} x$ .

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

Step 3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 3.2.4.2

Divide each term in $x \cdot 2^{\frac{3}{2}} = {(- 3)}^{\frac{3}{2}}$ by $2^{\frac{3}{2}}$ and simplify.

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Step 3.2.4.2.1

Divide each term in $x \cdot 2^{\frac{3}{2}} = {(- 3)}^{\frac{3}{2}}$ by $2^{\frac{3}{2}}$ .

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

Step 3.2.4.2.2

Simplify the left side.

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Step 3.2.4.2.2.1

Cancel the common factor.

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

Step 3.2.4.2.2.2

Divide $x$ by $1$ .

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

Step 3.2.4.3

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.2.4.4

Divide each term in $x \cdot 2^{\frac{3}{2}} = - {(- 3)}^{\frac{3}{2}}$ by $2^{\frac{3}{2}}$ and simplify.

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Step 3.2.4.4.1

Divide each term in $x \cdot 2^{\frac{3}{2}} = - {(- 3)}^{\frac{3}{2}}$ by $2^{\frac{3}{2}}$ .

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

Step 3.2.4.4.2

Simplify the left side.

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Step 3.2.4.4.2.1

Cancel the common factor.

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

Step 3.2.4.4.2.2

Divide $x$ by $1$ .

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

Step 3.2.4.4.3

Simplify the right side.

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Step 3.2.4.4.3.1

Move the negative in front of the fraction.

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

Step 3.2.4.5

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Add $2$ to both sides of the equation.

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

Step 4.2.2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

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

Step 4.2.3

Simplify the left side.

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Step 4.2.3.1

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

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Step 4.2.3.1.1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 4.2.3.1.1.1

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

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

Step 4.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.3.1.1.2.1

Cancel the common factor.

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

Step 4.2.3.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 2^{\frac{3}{2}}$

Step 4.2.3.1.1.3

Cancel the common factor of $3$ .

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Step 4.2.3.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 2^{\frac{3}{2}}$

Step 4.2.3.1.1.3.2

Rewrite the expression.

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

Step 4.2.3.1.2

Simplify.

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

Step 4.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 4.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 5

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

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

Step 6

Exclude the solutions that do not make $2 x^{\frac{4}{3}} - x^{\frac{2}{3}} - 6 = 0$ true.

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

Step 7