Solve by Factoring 3x^(4/3)+5x^(2/3)-2=0

$3 x^{\frac{4}{3}} + 5 x^{\frac{2}{3}} - 2 = 0$

Step 1

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

$3 {(x^{\frac{2}{3}})}^{2} + 5 x^{\frac{2}{3}} - 2 = 0$

Step 2

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

$3 u^{2} + 5 u - 2 = 0$

Step 3

Factor by grouping.

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 5$ .

Factor $5$ out of $5 u$ .

$3 u^{2} + 5 (u) - 2 = 0$

Rewrite $5$ as $- 1$ plus $6$

$3 u^{2} + (- 1 + 6) u - 2 = 0$

Apply the distributive property.

$3 u^{2} - 1 u + 6 u - 2 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 u^{2} - 1 u) + 6 u - 2 = 0$

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

$u (3 u - 1) + 2 (3 u - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 u - 1$ .

$(3 u - 1) (u + 2) = 0$

Step 4

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

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

Step 5

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

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

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

Step 6

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

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

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

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

Add $1$ to both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

Simplify the right side.

One to any power is one.

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

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

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

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

Divide each term in $(x \cdot 3^{\frac{3}{2}}) = 1$ by $3^{\frac{3}{2}}$ and simplify.

Divide each term in $(x \cdot 3^{\frac{3}{2}}) = 1$ by $3^{\frac{3}{2}}$ .

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

Simplify the left side.

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

Divide each term in $(x \cdot 3^{\frac{3}{2}}) = - 1$ by $3^{\frac{3}{2}}$ and simplify.

Divide each term in $(x \cdot 3^{\frac{3}{2}}) = - 1$ by $3^{\frac{3}{2}}$ .

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

Simplify the left side.

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

Step 7

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

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

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

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

Subtract $2$ from both sides of the equation.

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

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

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

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

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 8

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

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

Step 9

Exclude the solutions that do not make $(3 x^{\frac{2}{3}} - 1) (x^{\frac{2}{3}} + 2) = 0$ true.

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

Solve by Factoring 3x^(4/3)+5x^(2/3)-2=0

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