Algebra Examples

${(x^{2} - x + 4)}^{\frac{3}{2}} - 2 = 6$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Add $2$ to both sides of the equation.

${(x^{2} - x + 4)}^{\frac{3}{2}} = 6 + 2$

Step 1.2

Add $6$ and $2$ .

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

Step 2

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

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

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

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

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Step 3.1.1.1.1

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

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

Step 3.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.1.2.1

Cancel the common factor.

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

Step 3.1.1.1.2.2

Rewrite the expression.

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

Step 3.1.1.1.3

Cancel the common factor of $2$ .

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Step 3.1.1.1.3.1

Cancel the common factor.

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

Step 3.1.1.1.3.2

Rewrite the expression.

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

Step 3.1.1.2

Simplify.

$x^{2} - x + 4 = 8^{\frac{2}{3}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $8^{\frac{2}{3}}$ .

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Step 3.2.1.1

Simplify the expression.

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Step 3.2.1.1.1

Rewrite $8$ as $2^{3}$ .

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

Step 3.2.1.1.2

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

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

Step 3.2.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.2.1

Cancel the common factor.

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

Step 3.2.1.2.2

Rewrite the expression.

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

Step 3.2.1.3

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

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

Step 4

Solve for $x$ .

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Step 4.1

Subtract $4$ from both sides of the equation.

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

Step 4.2

Combine the opposite terms in $x^{2} - x + 4 - 4$ .

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Step 4.2.1

Subtract $4$ from $4$ .

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

Step 4.2.2

Add $x^{2} - x$ and $0$ .

$x^{2} - x = 0$

Step 4.3

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

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Step 4.3.1

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

$x \cdot x - x = 0$

Step 4.3.2

Factor $x$ out of $- x$ .

$x \cdot x + x \cdot - 1 = 0$

Step 4.3.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$x (x - 1) = 0$

Step 4.4

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

$x = 0$

$x - 1 = 0$

Step 4.5

Set $x$ equal to $0$ .

$x = 0$

Step 4.6

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

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Step 4.6.1

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

$x - 1 = 0$

Step 4.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.7

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

$x = 0, 1$