Solve by Factoring 9x^(2/3)-26x^(1/3)-3=0

$9 x^{\frac{2}{3}} - 26 x^{\frac{1}{3}} - 3 = 0$

Step 1

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

$9 {(x^{\frac{1}{3}})}^{2} - 26 x^{\frac{1}{3}} - 3 = 0$

Step 2

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

$9 u^{2} - 26 u - 3 = 0$

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 $a \cdot c = 9 \cdot - 3 = - 27$ and whose sum is $b = - 26$ .

Factor $- 26$ out of $- 26 u$ .

$9 u^{2} - 26 u - 3 = 0$

Rewrite $- 26$ as $1$ plus $- 27$

$9 u^{2} + (1 - 27) u - 3 = 0$

Apply the distributive property.

$9 u^{2} + 1 u - 27 u - 3 = 0$

Multiply $u$ by $1$ .

$9 u^{2} + u - 27 u - 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 u^{2} + u) - 27 u - 3 = 0$

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

$u (9 u + 1) - 3 (9 u + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $9 u + 1$ .

$(9 u + 1) (u - 3) = 0$

Step 4

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

$(9 x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} - 3) = 0$

Step 5

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

$9 x^{\frac{1}{3}} + 1 = 0$

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

Step 6

Set $9 x^{\frac{1}{3}} + 1$ equal to $0$ and solve for $x$ .

Set $9 x^{\frac{1}{3}} + 1$ equal to $0$ .

$9 x^{\frac{1}{3}} + 1 = 0$

Solve $9 x^{\frac{1}{3}} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$9 x^{\frac{1}{3}} = - 1$

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

${(9 x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${(9 x^{\frac{1}{3}})}^{3}$ .

Apply the product rule to $9 x^{\frac{1}{3}}$ .

$9^{3} {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Raise $9$ to the power of $3$ .

$729 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

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

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

$729 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$729 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Rewrite the expression.

$729 x^{1} = {(- 1)}^{3}$

Simplify.

$729 x = {(- 1)}^{3}$

Simplify the right side.

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

$729 x = - 1$

Divide each term in $729 x = - 1$ by $729$ and simplify.

Divide each term in $729 x = - 1$ by $729$ .

$\frac{729 x}{729} = \frac{- 1}{729}$

Simplify the left side.

Cancel the common factor of $729$ .

Cancel the common factor.

$\frac{729 x}{729} = \frac{- 1}{729}$

Divide $x$ by $1$ .

$x = \frac{- 1}{729}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{729}$

Step 7

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

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

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

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

Add $3$ to both sides of the equation.

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x = 3^{3}$

Simplify the right side.

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

$x = 27$

Step 8

The final solution is all the values that make $(9 x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} - 3) = 0$ true.

$x = - \frac{1}{729}, 27$

Solve by Factoring 9x^(2/3)-26x^(1/3)-3=0

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