Algebra Examples

$\sqrt{2 x + 4} = \sqrt{x} - 2$

Step 1

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

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

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 $\sqrt{2 x + 4}$ as ${(2 x + 4)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 x + 4)}^{1} = {(\sqrt{x} - 2)}^{2}$

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\sqrt{x} - 2)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(\sqrt{x} - 2)}^{2}$ as $(\sqrt{x} - 2) (\sqrt{x} - 2)$ .

$2 x + 4 = (\sqrt{x} - 2) (\sqrt{x} - 2)$

Step 2.3.1.2

Expand $(\sqrt{x} - 2) (\sqrt{x} - 2)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$2 x + 4 = \sqrt{x} (\sqrt{x} - 2) - 2 (\sqrt{x} - 2)$

Step 2.3.1.2.2

Apply the distributive property.

$2 x + 4 = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 2 - 2 (\sqrt{x} - 2)$

Step 2.3.1.2.3

Apply the distributive property.

$2 x + 4 = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 2 - 2 \sqrt{x} - 2 \cdot - 2$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 2.3.1.3.1.1.1

Raise $\sqrt{x}$ to the power of $1$ .

$2 x + 4 = {\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} \cdot - 2 - 2 \sqrt{x} - 2 \cdot - 2$

Step 2.3.1.3.1.1.2

Raise $\sqrt{x}$ to the power of $1$ .

$2 x + 4 = {\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} \cdot - 2 - 2 \sqrt{x} - 2 \cdot - 2$

Step 2.3.1.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x + 4 = {\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot - 2 - 2 \sqrt{x} - 2 \cdot - 2$

Step 2.3.1.3.1.1.4

Add $1$ and $1$ .

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

Step 2.3.1.3.1.2

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

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Step 2.3.1.3.1.2.1

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

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

Step 2.3.1.3.1.2.2

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

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

Step 2.3.1.3.1.2.3

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

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

Step 2.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.2.4.1

Cancel the common factor.

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

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

$2 x + 4 = x^{1} + \sqrt{x} \cdot - 2 - 2 \sqrt{x} - 2 \cdot - 2$

Step 2.3.1.3.1.2.5

Simplify.

$2 x + 4 = x + \sqrt{x} \cdot - 2 - 2 \sqrt{x} - 2 \cdot - 2$

Step 2.3.1.3.1.3

Move $- 2$ to the left of $\sqrt{x}$ .

$2 x + 4 = x - 2 \cdot \sqrt{x} - 2 \sqrt{x} - 2 \cdot - 2$

Step 2.3.1.3.1.4

Multiply $- 2$ by $- 2$ .

$2 x + 4 = x - 2 \sqrt{x} - 2 \sqrt{x} + 4$

Step 2.3.1.3.2

Subtract $2 \sqrt{x}$ from $- 2 \sqrt{x}$ .

$2 x + 4 = x - 4 \sqrt{x} + 4$

Step 3

Solve for $- 4 \sqrt{x}$ .

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Step 3.1

Rewrite the equation as $x - 4 \sqrt{x} + 4 = 2 x + 4$ .

$x - 4 \sqrt{x} + 4 = 2 x + 4$

Step 3.2

Move all terms not containing $- 4 \sqrt{x}$ to the right side of the equation.

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Step 3.2.1

Subtract $x$ from both sides of the equation.

$- 4 \sqrt{x} + 4 = 2 x + 4 - x$

Step 3.2.2

Subtract $4$ from both sides of the equation.

$- 4 \sqrt{x} = 2 x + 4 - x - 4$

Step 3.2.3

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

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Step 3.2.3.1

Subtract $4$ from $4$ .

$- 4 \sqrt{x} = 2 x - x + 0$

Step 3.2.3.2

Add $2 x - x$ and $0$ .

$- 4 \sqrt{x} = 2 x - x$

Step 3.2.4

Subtract $x$ from $2 x$ .

$- 4 \sqrt{x} = x$

Step 4

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

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

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

Apply the product rule to $- 4 x^{\frac{1}{2}}$ .

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

Step 5.2.1.2

Raise $- 4$ to the power of $2$ .

$16 {(x^{\frac{1}{2}})}^{2} = x^{2}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

$16 x^{\frac{1}{2} \cdot 2} = x^{2}$

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

$16 x^{\frac{1}{2} \cdot 2} = x^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$16 x^{1} = x^{2}$

Step 5.2.1.4

Simplify.

$16 x = x^{2}$

Step 6

Solve for $x$ .

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Step 6.1

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

$16 x - x^{2} = 0$

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 $x$ .

$16 u - u^{2} = 0$

Step 6.2.2

Factor $u$ out of $16 u - u^{2}$ .

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Step 6.2.2.1

Factor $u$ out of $16 u$ .

$u \cdot 16 - u^{2} = 0$

Step 6.2.2.2

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

$u \cdot 16 + u (- u) = 0$

Step 6.2.2.3

Factor $u$ out of $u \cdot 16 + u (- u)$ .

$u (16 - u) = 0$

Step 6.2.3

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

$x (16 - x) = 0$

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 $0$ .

$x = 0$

$16 - x = 0$

Step 6.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.5

Set $16 - x$ equal to $0$ and solve for $x$ .

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Step 6.5.1

Set $16 - x$ equal to $0$ .

$16 - x = 0$

Step 6.5.2

Solve $16 - x = 0$ for $x$ .

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Step 6.5.2.1

Subtract $16$ from both sides of the equation.

$- x = - 16$

Step 6.5.2.2

Divide each term in $- x = - 16$ by $- 1$ and simplify.

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Step 6.5.2.2.1

Divide each term in $- x = - 16$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 16}{- 1}$

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.

$\frac{x}{1} = \frac{- 16}{- 1}$

Step 6.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 16}{- 1}$

Step 6.5.2.2.3

Simplify the right side.

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Step 6.5.2.2.3.1

Divide $- 16$ by $- 1$ .

$x = 16$

Step 6.6

The final solution is all the values that make $x (16 - x) = 0$ true.

$x = 0, 16$

Step 7

Exclude the solutions that do not make $\sqrt{2 x + 4} = \sqrt{x} - 2$ true.

$No solution$