Solve for x 2x^(2/3)+7x^(1/3)-15=0

$2 x^{\frac{2}{3}} + 7 x^{\frac{1}{3}} - 15 = 0$

Step 1

Factor the left side of the equation.

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

$2 {(x^{\frac{1}{3}})}^{2} + 7 x^{\frac{1}{3}} - 15 = 0$

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

$2 u^{2} + 7 u - 15 = 0$

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 $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = 7$ .

Factor $7$ out of $7 u$ .

$2 u^{2} + 7 (u) - 15 = 0$

Rewrite $7$ as $- 3$ plus $10$

$2 u^{2} + (- 3 + 10) u - 15 = 0$

Apply the distributive property.

$2 u^{2} - 3 u + 10 u - 15 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 u^{2} - 3 u) + 10 u - 15 = 0$

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

$u (2 u - 3) + 5 (2 u - 3) = 0$

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

$(2 u - 3) (u + 5) = 0$

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

$(2 x^{\frac{1}{3}} - 3) (x^{\frac{1}{3}} + 5) = 0$

Step 2

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

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

$x^{\frac{1}{3}} + 5 = 0$

Step 3

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

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

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

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

Add $3$ to both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$8 x^{1} = 3^{3}$

Simplify.

$8 x = 3^{3}$

Simplify the right side.

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

$8 x = 27$

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

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

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

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 4

Set $x^{\frac{1}{3}} + 5$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{3}} + 5$ equal to $0$ .

$x^{\frac{1}{3}} + 5 = 0$

Solve $x^{\frac{1}{3}} + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$x^{\frac{1}{3}} = - 5$

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

${(x^{\frac{1}{3}})}^{3} = {(- 5)}^{3}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{3} \cdot 3} = {(- 5)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(- 5)}^{3}$

Rewrite the expression.

$x^{1} = {(- 5)}^{3}$

Simplify.

$x = {(- 5)}^{3}$

Simplify the right side.

Raise $- 5$ to the power of $3$ .

$x = - 125$

Step 5

The final solution is all the values that make $(2 x^{\frac{1}{3}} - 3) (x^{\frac{1}{3}} + 5) = 0$ true.

$x = \frac{27}{8}, - 125$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{27}{8}, - 125$

Decimal Form:

$x = 3.375, - 125$

Mixed Number Form:

$x = 3 \frac{3}{8}, - 125$

Solve for x 2x^(2/3)+7x^(1/3)-15=0

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