Algebra Examples

Solve $\sqrt{2 x + 21} = \sqrt{2 x} + 3$

Step 1

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

${\sqrt{2 x + 21}}^{2} = {(\sqrt{2 x} + 3)}^{2}$

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 $\sqrt{2 x + 21}$ as ${(2 x + 21)}^{\frac{1}{2}}$ .

${({(2 x + 21)}^{\frac{1}{2}})}^{2} = {(\sqrt{2 x} + 3)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 x + 21)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(2 x + 21)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(2 x + 21)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 x} + 3)}^{2}$

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.

${(2 x + 21)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 x} + 3)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 x + 21)}^{1} = {(\sqrt{2 x} + 3)}^{2}$

Step 2.2.1.2

Simplify.

$2 x + 21 = {(\sqrt{2 x} + 3)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\sqrt{2 x} + 3)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(\sqrt{2 x} + 3)}^{2}$ as $(\sqrt{2 x} + 3) (\sqrt{2 x} + 3)$ .

$2 x + 21 = (\sqrt{2 x} + 3) (\sqrt{2 x} + 3)$

Step 2.3.1.2

Expand $(\sqrt{2 x} + 3) (\sqrt{2 x} + 3)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$2 x + 21 = \sqrt{2 x} (\sqrt{2 x} + 3) + 3 (\sqrt{2 x} + 3)$

Step 2.3.1.2.2

Apply the distributive property.

$2 x + 21 = \sqrt{2 x} \sqrt{2 x} + \sqrt{2 x} \cdot 3 + 3 (\sqrt{2 x} + 3)$

Step 2.3.1.2.3

Apply the distributive property.

$2 x + 21 = \sqrt{2 x} \sqrt{2 x} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

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

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Step 2.3.1.3.1.1.1

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

$2 x + 21 = {\sqrt{2 x}}^{1} \sqrt{2 x} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.1.2

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

$2 x + 21 = {\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.1.3

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

$2 x + 21 = {\sqrt{2 x}}^{1 + 1} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.1.4

Add $1$ and $1$ .

$2 x + 21 = {\sqrt{2 x}}^{2} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.2

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

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Step 2.3.1.3.1.2.1

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

$2 x + 21 = {({(2 x)}^{\frac{1}{2}})}^{2} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.2.2

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

$2 x + 21 = {(2 x)}^{\frac{1}{2} \cdot 2} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.2.3

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

$2 x + 21 = {(2 x)}^{\frac{2}{2}} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

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.

$2 x + 21 = {(2 x)}^{\frac{2}{2}} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

$2 x + 21 = {(2 x)}^{1} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.2.5

Simplify.

$2 x + 21 = 2 x + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.3

Move $3$ to the left of $\sqrt{2 x}$ .

$2 x + 21 = 2 x + 3 \cdot \sqrt{2 x} + 3 \sqrt{2 x} + 3 \cdot 3$

Step 2.3.1.3.1.4

Multiply $3$ by $3$ .

$2 x + 21 = 2 x + 3 \sqrt{2 x} + 3 \sqrt{2 x} + 9$

Step 2.3.1.3.2

Add $3 \sqrt{2 x}$ and $3 \sqrt{2 x}$ .

$2 x + 21 = 2 x + 6 \sqrt{2 x} + 9$

Step 3

Solve for $6 \sqrt{2 x}$ .

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Step 3.1

Rewrite the equation as $2 x + 6 \sqrt{2 x} + 9 = 2 x + 21$ .

$2 x + 6 \sqrt{2 x} + 9 = 2 x + 21$

Step 3.2

Move all terms not containing $6 \sqrt{2 x}$ to the right side of the equation.

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Step 3.2.1

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

$6 \sqrt{2 x} + 9 = 2 x + 21 - 2 x$

Step 3.2.2

Subtract $9$ from both sides of the equation.

$6 \sqrt{2 x} = 2 x + 21 - 2 x - 9$

Step 3.2.3

Combine the opposite terms in $2 x + 21 - 2 x - 9$ .

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Step 3.2.3.1

Subtract $2 x$ from $2 x$ .

$6 \sqrt{2 x} = 0 + 21 - 9$

Step 3.2.3.2

Add $0$ and $21$ .

$6 \sqrt{2 x} = 21 - 9$

Step 3.2.4

Subtract $9$ from $21$ .

$6 \sqrt{2 x} = 12$

Step 4

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

${(6 \sqrt{2 x})}^{2} = 12^{2}$

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

${(6 {(2 x)}^{\frac{1}{2}})}^{2} = 12^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

Apply the product rule to $2 x$ .

${(6 (2^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} = 12^{2}$

Step 5.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.2.1

Apply the product rule to $6 \cdot 2^{\frac{1}{2}} x^{\frac{1}{2}}$ .

${(6 \cdot 2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 5.2.1.2.2

Apply the product rule to $6 \cdot 2^{\frac{1}{2}}$ .

$6^{2} \cdot {(2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 5.2.1.3

Raise $6$ to the power of $2$ .

$36 \cdot {(2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 5.2.1.4

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

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Step 5.2.1.4.1

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

$36 \cdot 2^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 5.2.1.4.2

Cancel the common factor of $2$ .

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Step 5.2.1.4.2.1

Cancel the common factor.

$36 \cdot 2^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 5.2.1.4.2.2

Rewrite the expression.

$36 \cdot 2^{1} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 5.2.1.5

Evaluate the exponent.

$36 \cdot 2 {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 5.2.1.6

Multiply $36$ by $2$ .

$72 {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 5.2.1.7

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

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Step 5.2.1.7.1

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

$72 x^{\frac{1}{2} \cdot 2} = 12^{2}$

Step 5.2.1.7.2

Cancel the common factor of $2$ .

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Step 5.2.1.7.2.1

Cancel the common factor.

$72 x^{\frac{1}{2} \cdot 2} = 12^{2}$

Step 5.2.1.7.2.2

Rewrite the expression.

$72 x^{1} = 12^{2}$

Step 5.2.1.8

Simplify.

$72 x = 12^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Raise $12$ to the power of $2$ .

$72 x = 144$

Step 6

Divide each term in $72 x = 144$ by $72$ and simplify.

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Step 6.1

Divide each term in $72 x = 144$ by $72$ .

$\frac{72 x}{72} = \frac{144}{72}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $72$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{72 x}{72} = \frac{144}{72}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{144}{72}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $144$ by $72$ .

$x = 2$