Algebra Examples

$\sqrt{x^{2} + 6 x} = x + \sqrt{2 x}$

Step 1

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

${\sqrt{x^{2} + 6 x}}^{2} = {(x + \sqrt{2 x})}^{2}$

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 $\sqrt{x^{2} + 6 x}$ as ${(x^{2} + 6 x)}^{\frac{1}{2}}$ .

${({(x^{2} + 6 x)}^{\frac{1}{2}})}^{2} = {(x + \sqrt{2 x})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x^{2} + 6 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x^{2} + 6 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x^{2} + 6 x)}^{\frac{1}{2} \cdot 2} = {(x + \sqrt{2 x})}^{2}$

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.

${(x^{2} + 6 x)}^{\frac{1}{2} \cdot 2} = {(x + \sqrt{2 x})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 6 x)}^{1} = {(x + \sqrt{2 x})}^{2}$

Step 2.2.1.2

Simplify.

$x^{2} + 6 x = {(x + \sqrt{2 x})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

$x^{2} + 6 x = (x + \sqrt{2 x}) (x + \sqrt{2 x})$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

$x^{2} + 6 x = x (x + \sqrt{2 x}) + \sqrt{2 x} (x + \sqrt{2 x})$

Step 2.3.1.2.2

Apply the distributive property.

$x^{2} + 6 x = x \cdot x + x \sqrt{2 x} + \sqrt{2 x} (x + \sqrt{2 x})$

Step 2.3.1.2.3

Apply the distributive property.

$x^{2} + 6 x = x \cdot x + x \sqrt{2 x} + \sqrt{2 x} x + \sqrt{2 x} \sqrt{2 x}$

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 $x$ by $x$ .

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + \sqrt{2 x} \sqrt{2 x}$

Step 2.3.1.3.1.2

Multiply $\sqrt{2 x} \sqrt{2 x}$ .

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Step 2.3.1.3.1.2.1

Raise $\sqrt{2 x}$ to the power of $1$ .

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + {\sqrt{2 x}}^{1} \sqrt{2 x}$

Step 2.3.1.3.1.2.2

Raise $\sqrt{2 x}$ to the power of $1$ .

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + {\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1}$

Step 2.3.1.3.1.2.3

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

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + {\sqrt{2 x}}^{1 + 1}$

Step 2.3.1.3.1.2.4

Add $1$ and $1$ .

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + {\sqrt{2 x}}^{2}$

Step 2.3.1.3.1.3

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

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Step 2.3.1.3.1.3.1

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

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + {({(2 x)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.3.1.3.2

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

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + {(2 x)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3.1.3.3

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

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + {(2 x)}^{\frac{2}{2}}$

Step 2.3.1.3.1.3.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.3.4.1

Cancel the common factor.

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + {(2 x)}^{\frac{2}{2}}$

Step 2.3.1.3.1.3.4.2

Rewrite the expression.

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + {(2 x)}^{1}$

Step 2.3.1.3.1.3.5

Simplify.

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + \sqrt{2 x} x + 2 x$

Step 2.3.1.3.2

Reorder the factors of $\sqrt{2 x} x$ .

$x^{2} + 6 x = x^{2} + x \sqrt{2 x} + x \sqrt{2 x} + 2 x$

Step 2.3.1.3.3

Add $x \sqrt{2 x}$ and $x \sqrt{2 x}$ .

$x^{2} + 6 x = x^{2} + 2 x \sqrt{2 x} + 2 x$

Step 3

Solve for $2 x \sqrt{2 x}$ .

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Step 3.1

Rewrite the equation as $x^{2} + 2 x \sqrt{2 x} + 2 x = x^{2} + 6 x$ .

$x^{2} + 2 x \sqrt{2 x} + 2 x = x^{2} + 6 x$

Step 3.2

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

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Step 3.2.1

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

$2 x \sqrt{2 x} + 2 x = x^{2} + 6 x - x^{2}$

Step 3.2.2

Subtract $2 x$ from both sides of the equation.

$2 x \sqrt{2 x} = x^{2} + 6 x - x^{2} - 2 x$

Step 3.2.3

Combine the opposite terms in $x^{2} + 6 x - x^{2} - 2 x$ .

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Step 3.2.3.1

Subtract $x^{2}$ from $x^{2}$ .

$2 x \sqrt{2 x} = 6 x + 0 - 2 x$

Step 3.2.3.2

Add $6 x$ and $0$ .

$2 x \sqrt{2 x} = 6 x - 2 x$

Step 3.2.4

Subtract $2 x$ from $6 x$ .

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

Step 4

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

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

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

Apply the product rule to $2 x$ .

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

Step 5.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.3

Multiply $2$ by $2^{\frac{1}{2}}$ by adding the exponents.

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Step 5.2.1.3.1

Multiply $2$ by $2^{\frac{1}{2}}$ .

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Step 5.2.1.3.1.1

Raise $2$ to the power of $1$ .

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

Step 5.2.1.3.1.2

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

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

Step 5.2.1.3.2

Write $1$ as a fraction with a common denominator.

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

Step 5.2.1.3.3

Combine the numerators over the common denominator.

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

Step 5.2.1.3.4

Add $2$ and $1$ .

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

Step 5.2.1.4

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 5.2.1.4.1

Move $x^{\frac{1}{2}}$ .

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

Step 5.2.1.4.2

Multiply $x^{\frac{1}{2}}$ by $x$ .

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Step 5.2.1.4.2.1

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

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

Step 5.2.1.4.2.2

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

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

Step 5.2.1.4.3

Write $1$ as a fraction with a common denominator.

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

Step 5.2.1.4.4

Combine the numerators over the common denominator.

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

Step 5.2.1.4.5

Add $1$ and $2$ .

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

Step 5.2.1.5

Apply the product rule to $2^{\frac{3}{2}} x^{\frac{3}{2}}$ .

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

Step 5.2.1.6

Multiply the exponents in ${(2^{\frac{3}{2}})}^{2}$ .

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Step 5.2.1.6.1

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

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

Step 5.2.1.6.2

Cancel the common factor of $2$ .

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Step 5.2.1.6.2.1

Cancel the common factor.

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

Step 5.2.1.6.2.2

Rewrite the expression.

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

Step 5.2.1.7

Raise $2$ to the power of $3$ .

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

Step 5.2.1.8

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

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Step 5.2.1.8.1

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

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

Step 5.2.1.8.2

Cancel the common factor of $2$ .

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Step 5.2.1.8.2.1

Cancel the common factor.

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

Step 5.2.1.8.2.2

Rewrite the expression.

$8 x^{3} = {(4 x)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(4 x)}^{2}$ .

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Step 5.3.1.1

Apply the product rule to $4 x$ .

$8 x^{3} = 4^{2} x^{2}$

Step 5.3.1.2

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

$8 x^{3} = 16 x^{2}$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $16 x^{2}$ from both sides of the equation.

$8 x^{3} - 16 x^{2} = 0$

Step 6.2

Factor $8 x^{2}$ out of $8 x^{3} - 16 x^{2}$ .

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Step 6.2.1

Factor $8 x^{2}$ out of $8 x^{3}$ .

$8 x^{2} (x) - 16 x^{2} = 0$

Step 6.2.2

Factor $8 x^{2}$ out of $- 16 x^{2}$ .

$8 x^{2} (x) + 8 x^{2} (- 2) = 0$

Step 6.2.3

Factor $8 x^{2}$ out of $8 x^{2} (x) + 8 x^{2} (- 2)$ .

$8 x^{2} (x - 2) = 0$

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 $0$ .

$x^{2} = 0$

$x - 2 = 0$

Step 6.4

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

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Step 6.4.1

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

$x^{2} = 0$

Step 6.4.2

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

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Step 6.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 6.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 6.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 6.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.5

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

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Step 6.5.1

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

$x - 2 = 0$

Step 6.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6.6

The final solution is all the values that make $8 x^{2} (x - 2) = 0$ true.

$x = 0, 2$