Solve for x square root of 2x+5 = square root of x+2+1

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

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

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

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 5}$ as ${(2 x + 5)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

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

Multiply $1$ by $1$ .

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

Add $2$ and $1$ .

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

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

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

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

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

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $3$ from both sides of the equation.

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

Subtract $x$ from $2 x$ .

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

Subtract $3$ from $5$ .

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $4$ by $2$ .

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $2$ by $2$ .

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

Add $2 x$ and $2 x$ .

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

Solve for $x$ .

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

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

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

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

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

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

Subtract $4 x$ from $4 x$ .

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

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

$x^{2} + 4 = 8$

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

Subtract $4$ from both sides of the equation.

$x^{2} = 8 - 4$

Subtract $4$ from $8$ .

$x^{2} = 4$

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

$x = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

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

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

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

$x = \pm 2$

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

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

$x = 2$

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

$x = - 2$

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

$x = 2, - 2$

Solve for x square root of 2x+5 = square root of x+2+1

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