Algebra Examples

$\sqrt[3]{5 x^{2} + 8 x - 4} - \sqrt[3]{9 x} = 0$

Step 1

Add $\sqrt[3]{9 x}$ to both sides of the equation.

$\sqrt[3]{5 x^{2} + 8 x - 4} = \sqrt[3]{9 x}$

Step 2

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{5 x^{2} + 8 x - 4}}^{3} = {\sqrt[3]{9 x}}^{3}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 x^{2} + 8 x - 4}$ as ${(5 x^{2} + 8 x - 4)}^{\frac{1}{3}}$ .

${({(5 x^{2} + 8 x - 4)}^{\frac{1}{3}})}^{3} = {\sqrt[3]{9 x}}^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(5 x^{2} + 8 x - 4)}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(5 x^{2} + 8 x - 4)}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.1.1

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

${(5 x^{2} + 8 x - 4)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{9 x}}^{3}$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(5 x^{2} + 8 x - 4)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{9 x}}^{3}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(5 x^{2} + 8 x - 4)}^{1} = {\sqrt[3]{9 x}}^{3}$

Step 3.2.1.2

Simplify.

$5 x^{2} + 8 x - 4 = {\sqrt[3]{9 x}}^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Rewrite ${\sqrt[3]{9 x}}^{3}$ as $9 x$ .

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Step 3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 x}$ as ${(9 x)}^{\frac{1}{3}}$ .

$5 x^{2} + 8 x - 4 = {({(9 x)}^{\frac{1}{3}})}^{3}$

Step 3.3.1.2

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

$5 x^{2} + 8 x - 4 = {(9 x)}^{\frac{1}{3} \cdot 3}$

Step 3.3.1.3

Combine $\frac{1}{3}$ and $3$ .

$5 x^{2} + 8 x - 4 = {(9 x)}^{\frac{3}{3}}$

Step 3.3.1.4

Cancel the common factor of $3$ .

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Step 3.3.1.4.1

Cancel the common factor.

$5 x^{2} + 8 x - 4 = {(9 x)}^{\frac{3}{3}}$

Step 3.3.1.4.2

Rewrite the expression.

$5 x^{2} + 8 x - 4 = {(9 x)}^{1}$

Step 3.3.1.5

Simplify.

$5 x^{2} + 8 x - 4 = 9 x$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $9 x$ from both sides of the equation.

$5 x^{2} + 8 x - 4 - 9 x = 0$

Step 4.1.2

Subtract $9 x$ from $8 x$ .

$5 x^{2} - x - 4 = 0$

Step 4.2

Factor by grouping.

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Step 4.2.1

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 $a \cdot c = 5 \cdot - 4 = - 20$ and whose sum is $b = - 1$ .

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Step 4.2.1.1

Factor $- 1$ out of $- x$ .

$5 x^{2} - (x) - 4 = 0$

Step 4.2.1.2

Rewrite $- 1$ as $4$ plus $- 5$

$5 x^{2} + (4 - 5) x - 4 = 0$

Step 4.2.1.3

Apply the distributive property.

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

Step 4.2.2

Factor out the greatest common factor from each group.

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Step 4.2.2.1

Group the first two terms and the last two terms.

$(5 x^{2} + 4 x) - 5 x - 4 = 0$

Step 4.2.2.2

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

$x (5 x + 4) - (5 x + 4) = 0$

Step 4.2.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 4$ .

$(5 x + 4) (x - 1) = 0$

Step 4.3

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

$5 x + 4 = 0$

$x - 1 = 0$

Step 4.4

Set $5 x + 4$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $5 x + 4$ equal to $0$ .

$5 x + 4 = 0$

Step 4.4.2

Solve $5 x + 4 = 0$ for $x$ .

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Step 4.4.2.1

Subtract $4$ from both sides of the equation.

$5 x = - 4$

Step 4.4.2.2

Divide each term in $5 x = - 4$ by $5$ and simplify.

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Step 4.4.2.2.1

Divide each term in $5 x = - 4$ by $5$ .

$\frac{5 x}{5} = \frac{- 4}{5}$

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 4}{5}$

Step 4.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{5}$

Step 4.4.2.2.3

Simplify the right side.

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Step 4.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{5}$

Step 4.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.6

The final solution is all the values that make $(5 x + 4) (x - 1) = 0$ true.

$x = - \frac{4}{5}, 1$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4}{5}, 1$

Decimal Form:

$x = - 0.8, 1$