Algebra Examples

$\sqrt{2 x + 5} = \sqrt{x + 2} + 1$

Step 1

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

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

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 $\sqrt{2 x + 5}$ as ${(2 x + 5)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 x + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Expand $(\sqrt{x + 2} + 1) (\sqrt{x + 2} + 1)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Apply the distributive property.

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

Step 2.3.1.2.3

Apply the distributive property.

$2 x + 5 = \sqrt{x + 2} \sqrt{x + 2} + \sqrt{x + 2} \cdot 1 + 1 \sqrt{x + 2} + 1 \cdot 1$

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

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Step 2.3.1.3.1.1.1

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

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

Step 2.3.1.3.1.1.2

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

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

Step 2.3.1.3.1.1.3

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

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

Step 2.3.1.3.1.1.4

Add $1$ and $1$ .

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

Step 2.3.1.3.1.2

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

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Step 2.3.1.3.1.2.1

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

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

Step 2.3.1.3.1.2.2

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

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

Step 2.3.1.3.1.2.3

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

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

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.

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

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

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

Step 2.3.1.3.1.2.5

Simplify.

$2 x + 5 = x + 2 + \sqrt{x + 2} \cdot 1 + 1 \sqrt{x + 2} + 1 \cdot 1$

Step 2.3.1.3.1.3

Multiply $\sqrt{x + 2}$ by $1$ .

$2 x + 5 = x + 2 + \sqrt{x + 2} + 1 \sqrt{x + 2} + 1 \cdot 1$

Step 2.3.1.3.1.4

Multiply $\sqrt{x + 2}$ by $1$ .

$2 x + 5 = x + 2 + \sqrt{x + 2} + \sqrt{x + 2} + 1 \cdot 1$

Step 2.3.1.3.1.5

Multiply $1$ by $1$ .

$2 x + 5 = x + 2 + \sqrt{x + 2} + \sqrt{x + 2} + 1$

Step 2.3.1.3.2

Add $2$ and $1$ .

$2 x + 5 = x + 3 + \sqrt{x + 2} + \sqrt{x + 2}$

Step 2.3.1.3.3

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

$2 x + 5 = x + 3 + 2 \sqrt{x + 2}$

Step 3

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

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Step 3.1

Rewrite the equation as $x + 3 + 2 \sqrt{x + 2} = 2 x + 5$ .

$x + 3 + 2 \sqrt{x + 2} = 2 x + 5$

Step 3.2

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

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Step 3.2.1

Subtract $x$ from both sides of the equation.

$3 + 2 \sqrt{x + 2} = 2 x + 5 - x$

Step 3.2.2

Subtract $3$ from both sides of the equation.

$2 \sqrt{x + 2} = 2 x + 5 - x - 3$

Step 3.2.3

Subtract $x$ from $2 x$ .

$2 \sqrt{x + 2} = x + 5 - 3$

Step 3.2.4

Subtract $3$ from $5$ .

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

Step 4

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

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

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

Apply the product rule to $2 {(x + 2)}^{\frac{1}{2}}$ .

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

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

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.

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

Step 5.2.1.3.2.2

Rewrite the expression.

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

Step 5.2.1.4

Simplify.

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

Step 5.2.1.5

Apply the distributive property.

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

Step 5.2.1.6

Multiply $4$ by $2$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

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

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Step 5.3.1.1

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

$4 x + 8 = (x + 2) (x + 2)$

Step 5.3.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 5.3.1.2.1

Apply the distributive property.

$4 x + 8 = x (x + 2) + 2 (x + 2)$

Step 5.3.1.2.2

Apply the distributive property.

$4 x + 8 = x \cdot x + x \cdot 2 + 2 (x + 2)$

Step 5.3.1.2.3

Apply the distributive property.

$4 x + 8 = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

Step 5.3.1.3

Simplify and combine like terms.

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Step 5.3.1.3.1

Simplify each term.

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Step 5.3.1.3.1.1

Multiply $x$ by $x$ .

$4 x + 8 = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2$

Step 5.3.1.3.1.2

Move $2$ to the left of $x$ .

$4 x + 8 = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2$

Step 5.3.1.3.1.3

Multiply $2$ by $2$ .

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

Step 5.3.1.3.2

Add $2 x$ and $2 x$ .

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

Step 6

Solve for $x$ .

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

Subtract $4 x$ from both sides of the equation.

$x^{2} + 4 x + 4 - 4 x = 8$

Step 6.2.2

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

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Step 6.2.2.1

Subtract $4 x$ from $4 x$ .

$x^{2} + 0 + 4 = 8$

Step 6.2.2.2

Add $x^{2}$ and $0$ .

$x^{2} + 4 = 8$

Step 6.3

Move all terms not containing $x$ to the right side of the equation.

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Step 6.3.1

Subtract $4$ from both sides of the equation.

$x^{2} = 8 - 4$

Step 6.3.2

Subtract $4$ from $8$ .

$x^{2} = 4$

Step 6.4

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

$x = \pm \sqrt{4}$

Step 6.5

Simplify $\pm \sqrt{4}$ .

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Step 6.5.1

Rewrite $4$ as $2^{2}$ .

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

Step 6.5.2

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

$x = \pm 2$

Step 6.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 6.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 6.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$