Find All Complex Solutions 3x^(2/3)+x^(1/3)-2=0

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

Step 1

Factor the left side of the equation.

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

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

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

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

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 $b = 1$ .

Multiply by $1$ .

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

Rewrite $1$ as $- 2$ plus $3$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

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

Step 2

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

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

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

Step 3

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

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

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

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

Add $2$ to both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

$27 {(x^{\frac{1}{3}})}^{3} = 2^{3}$

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

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

$27 x^{\frac{1}{3} \cdot 3} = 2^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$27 x^{\frac{1}{3} \cdot 3} = 2^{3}$

Rewrite the expression.

$27 x^{1} = 2^{3}$

Simplify.

$27 x = 2^{3}$

Simplify the right side.

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

$27 x = 8$

Divide each term in $27 x = 8$ by $27$ and simplify.

Divide each term in $27 x = 8$ by $27$ .

$\frac{27 x}{27} = \frac{8}{27}$

Simplify the left side.

Cancel the common factor of $27$ .

Cancel the common factor.

$\frac{27 x}{27} = \frac{8}{27}$

Divide $x$ by $1$ .

$x = \frac{8}{27}$

Step 4

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

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

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

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

Subtract $1$ from both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$x = - 1$

Step 5

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

$x = \frac{8}{27}, - 1$

Find All Complex Solutions 3x^(2/3)+x^(1/3)-2=0

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