Algebra Examples

$\sqrt{{(5 + x^{2})}^{2}} = 5 + x^{2}$

Step 1

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

${\sqrt{{(5 + x^{2})}^{2}}}^{2} = {(5 + x^{2})}^{2}$

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 $\sqrt{{(5 + x^{2})}^{2}}$ as ${(5 + x^{2})}^{\frac{2}{2}}$ .

${({(5 + x^{2})}^{\frac{2}{2}})}^{2} = {(5 + x^{2})}^{2}$

Step 2.2

Divide $2$ by $2$ .

${({(5 + x^{2})}^{1})}^{2} = {(5 + x^{2})}^{2}$

Step 2.3

Simplify the left side.

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Step 2.3.1

Multiply the exponents in ${({(5 + x^{2})}^{1})}^{2}$ .

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Step 2.3.1.1

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

${(5 + x^{2})}^{1 \cdot 2} = {(5 + x^{2})}^{2}$

Step 2.3.1.2

Multiply $2$ by $1$ .

${(5 + x^{2})}^{2} = {(5 + x^{2})}^{2}$

Step 2.4

Simplify the right side.

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Step 2.4.1

Simplify ${(5 + x^{2})}^{2}$ .

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Step 2.4.1.1

Rewrite ${(5 + x^{2})}^{2}$ as $(5 + x^{2}) (5 + x^{2})$ .

${(5 + x^{2})}^{2} = (5 + x^{2}) (5 + x^{2})$

Step 2.4.1.2

Expand $(5 + x^{2}) (5 + x^{2})$ using the FOIL Method.

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Step 2.4.1.2.1

Apply the distributive property.

${(5 + x^{2})}^{2} = 5 (5 + x^{2}) + x^{2} (5 + x^{2})$

Step 2.4.1.2.2

Apply the distributive property.

${(5 + x^{2})}^{2} = 5 \cdot 5 + 5 x^{2} + x^{2} (5 + x^{2})$

Step 2.4.1.2.3

Apply the distributive property.

${(5 + x^{2})}^{2} = 5 \cdot 5 + 5 x^{2} + x^{2} \cdot 5 + x^{2} x^{2}$

Step 2.4.1.3

Simplify and combine like terms.

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Step 2.4.1.3.1

Simplify each term.

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Step 2.4.1.3.1.1

Multiply $5$ by $5$ .

${(5 + x^{2})}^{2} = 25 + 5 x^{2} + x^{2} \cdot 5 + x^{2} x^{2}$

Step 2.4.1.3.1.2

Move $5$ to the left of $x^{2}$ .

${(5 + x^{2})}^{2} = 25 + 5 x^{2} + 5 \cdot x^{2} + x^{2} x^{2}$

Step 2.4.1.3.1.3

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

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Step 2.4.1.3.1.3.1

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

${(5 + x^{2})}^{2} = 25 + 5 x^{2} + 5 x^{2} + x^{2 + 2}$

Step 2.4.1.3.1.3.2

Add $2$ and $2$ .

${(5 + x^{2})}^{2} = 25 + 5 x^{2} + 5 x^{2} + x^{4}$

Step 2.4.1.3.2

Add $5 x^{2}$ and $5 x^{2}$ .

${(5 + x^{2})}^{2} = 25 + 10 x^{2} + x^{4}$

Step 3

Solve for $x$ .

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Step 3.1

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

$25 + 10 x^{2} + x^{4} = {(5 + x^{2})}^{2}$

Step 3.2

Simplify ${(5 + x^{2})}^{2}$ .

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Step 3.2.1

Rewrite.

$25 + 10 x^{2} + x^{4} = 0 + 0 + {(5 + x^{2})}^{2}$

Step 3.2.2

Rewrite ${(5 + x^{2})}^{2}$ as $(5 + x^{2}) (5 + x^{2})$ .

$25 + 10 x^{2} + x^{4} = (5 + x^{2}) (5 + x^{2})$

Step 3.2.3

Expand $(5 + x^{2}) (5 + x^{2})$ using the FOIL Method.

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Step 3.2.3.1

Apply the distributive property.

$25 + 10 x^{2} + x^{4} = 5 (5 + x^{2}) + x^{2} (5 + x^{2})$

Step 3.2.3.2

Apply the distributive property.

$25 + 10 x^{2} + x^{4} = 5 \cdot 5 + 5 x^{2} + x^{2} (5 + x^{2})$

Step 3.2.3.3

Apply the distributive property.

$25 + 10 x^{2} + x^{4} = 5 \cdot 5 + 5 x^{2} + x^{2} \cdot 5 + x^{2} x^{2}$

Step 3.2.4

Simplify and combine like terms.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Multiply $5$ by $5$ .

$25 + 10 x^{2} + x^{4} = 25 + 5 x^{2} + x^{2} \cdot 5 + x^{2} x^{2}$

Step 3.2.4.1.2

Move $5$ to the left of $x^{2}$ .

$25 + 10 x^{2} + x^{4} = 25 + 5 x^{2} + 5 \cdot x^{2} + x^{2} x^{2}$

Step 3.2.4.1.3

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

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Step 3.2.4.1.3.1

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

$25 + 10 x^{2} + x^{4} = 25 + 5 x^{2} + 5 x^{2} + x^{2 + 2}$

Step 3.2.4.1.3.2

Add $2$ and $2$ .

$25 + 10 x^{2} + x^{4} = 25 + 5 x^{2} + 5 x^{2} + x^{4}$

Step 3.2.4.2

Add $5 x^{2}$ and $5 x^{2}$ .

$25 + 10 x^{2} + x^{4} = 25 + 10 x^{2} + x^{4}$

Step 3.3

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.1

Subtract $10 x^{2}$ from both sides of the equation.

$25 + 10 x^{2} + x^{4} - 10 x^{2} = 25 + x^{4}$

Step 3.3.2

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

$25 + 10 x^{2} + x^{4} - 10 x^{2} - x^{4} = 25$

Step 3.3.3

Combine the opposite terms in $25 + 10 x^{2} + x^{4} - 10 x^{2} - x^{4}$ .

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Step 3.3.3.1

Subtract $10 x^{2}$ from $10 x^{2}$ .

$25 + x^{4} + 0 - x^{4} = 25$

Step 3.3.3.2

Add $25 + x^{4}$ and $0$ .

$25 + x^{4} - x^{4} = 25$

Step 3.3.3.3

Subtract $x^{4}$ from $x^{4}$ .

$25 + 0 = 25$

Step 3.3.3.4

Add $25$ and $0$ .

$25 = 25$

Step 3.4

Since $25 = 25$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$