Solve for x cube root of 4x^2-4x=x

$\sqrt[3]{4 x^{2} - 4 x} = x$

Step 1

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

${\sqrt[3]{4 x^{2} - 4 x}}^{3} = x^{3}$

Step 2

Simplify each side of the equation.

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

${({(4 x^{2} - 4 x)}^{\frac{1}{3}})}^{3} = x^{3}$

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

${(4 x^{2} - 4 x)}^{1} = x^{3}$

Simplify.

$4 x^{2} - 4 x = x^{3}$

Step 3

Solve for $x$ .

Subtract $x^{3}$ from both sides of the equation.

$4 x^{2} - 4 x - x^{3} = 0$

Factor the left side of the equation.

Factor $- x$ out of $4 x^{2} - 4 x - x^{3}$ .

Reorder the expression.

Move $- 4 x$ .

$4 x^{2} - x^{3} - 4 x = 0$

Reorder $4 x^{2}$ and $- x^{3}$ .

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

Factor $- x$ out of $- x^{3}$ .

$- x \cdot x^{2} + 4 x^{2} - 4 x = 0$

Factor $- x$ out of $4 x^{2}$ .

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

Factor $- x$ out of $- 4 x$ .

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

Factor $- x$ out of $- x (x^{2}) - x (- 4 x)$ .

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

Factor $- x$ out of $- x (x^{2} - 4 x) - x (4)$ .

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

Factor using the perfect square rule.

Rewrite $4$ as $2^{2}$ .

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

$- x (x^{2} - 2 \cdot x \cdot 2 + 2^{2}) = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

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

$x = 0$

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

Set $x$ equal to $0$ .

$x = 0$

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{2}$ equal to $0$ .

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

Solve ${(x - 2)}^{2} = 0$ for $x$ .

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

$x = 0, 2$

Solve for x cube root of 4x^2-4x=x

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