Algebra Examples

$x^{2} = \sqrt{x}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{x} = x^{2}$

Step 2

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

${\sqrt{x}}^{2} = {(x^{2})}^{2}$

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(x^{2})}^{2}$

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.3.1.1

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

$x = x^{2 \cdot 2}$

Step 3.3.1.2

Multiply $2$ by $2$ .

$x = x^{4}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $x^{4}$ from both sides of the equation.

$x - x^{4} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

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

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Step 4.2.1.1

Raise $x$ to the power of $1$ .

$x - x^{4} = 0$

Step 4.2.1.2

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - x^{4} = 0$

Step 4.2.1.3

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

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

Step 4.2.1.4

Factor $x$ out of $x \cdot 1 + x (- x^{3})$ .

$x (1 - x^{3}) = 0$

Step 4.2.2

Rewrite $1$ as $1^{3}$ .

$x (1^{3} - x^{3}) = 0$

Step 4.2.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = x$ .

$x ((1 - x) (1^{2} + 1 x + x^{2})) = 0$

Step 4.2.4

Factor.

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Step 4.2.4.1

Simplify.

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Step 4.2.4.1.1

One to any power is one.

$x ((1 - x) (1 + 1 x + x^{2})) = 0$

Step 4.2.4.1.2

Multiply $x$ by $1$ .

$x ((1 - x) (1 + x + x^{2})) = 0$

Step 4.2.4.2

Remove unnecessary parentheses.

$x (1 - x) (1 + x + x^{2}) = 0$

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 $0$ .

$x = 0$

$1 - x = 0$

$1 + x + x^{2} = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

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

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Step 4.5.1

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

$1 - x = 0$

Step 4.5.2

Solve $1 - x = 0$ for $x$ .

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Step 4.5.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 4.5.2.2

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

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Step 4.5.2.2.1

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

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

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 4.5.2.2.2.2

Divide $x$ by $1$ .

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

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 4.6

Set $1 + x + x^{2}$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $1 + x + x^{2}$ equal to $0$ .

$1 + x + x^{2} = 0$

Step 4.6.2

Solve $1 + x + x^{2} = 0$ for $x$ .

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Step 4.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.6.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 4.6.2.3

Simplify.

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Step 4.6.2.3.1

Simplify the numerator.

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Step 4.6.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 4.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 4.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 4.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 4.6.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 4.6.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 4.6.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 4.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 4.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 4.6.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 4.7

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

$x = 0, 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$