Algebra Examples

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + \sqrt{2 x})}^{1} = {\sqrt{2 x}}^{2}$

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

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

Step 2.3.1.4.2

Rewrite the expression.

$x + \sqrt{2 x} = {(2 x)}^{1}$

Step 2.3.1.5

Simplify.

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

Step 3

Move all terms not containing $\sqrt{2 x}$ to the right side of the equation.

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Step 3.1

Subtract $x$ from both sides of the equation.

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

Step 3.2

Subtract $x$ from $2 x$ .

$\sqrt{2 x} = x$

Step 4

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

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

Step 5

Simplify each side of the equation.

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

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

Step 5.2.1.2

Simplify.

$2 x = x^{2}$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$2 u - u^{2} = 0$

Step 6.2.2

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

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Step 6.2.2.1

Factor $u$ out of $2 u$ .

$u \cdot 2 - u^{2} = 0$

Step 6.2.2.2

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

$u \cdot 2 + u (- u) = 0$

Step 6.2.2.3

Factor $u$ out of $u \cdot 2 + u (- u)$ .

$u (2 - u) = 0$

Step 6.2.3

Replace all occurrences of $u$ with $x$ .

$x (2 - x) = 0$

Step 6.3

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

$x = 0$

$2 - x = 0$

Step 6.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.5

Set $2 - x$ equal to $0$ and solve for $x$ .

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Step 6.5.1

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

$2 - x = 0$

Step 6.5.2

Solve $2 - x = 0$ for $x$ .

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Step 6.5.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 6.5.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 6.5.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 6.5.2.2.2

Simplify the left side.

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Step 6.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 6.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 6.5.2.2.3

Simplify the right side.

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Step 6.5.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 6.6

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

$x = 0, 2$