Solve for x 2+ square root of x+1- square root of x+5=0

$2 + \sqrt{x + 1} - \sqrt{x + 5} = 0$

Step 1

Move all terms not containing $\sqrt{x + 1}$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

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

Add $\sqrt{x + 5}$ to both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

Expand $(- 2 + \sqrt{x + 5}) (- 2 + \sqrt{x + 5})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $- 2$ .

$x + 1 = 4 - 2 \sqrt{x + 5} + \sqrt{x + 5} \cdot - 2 + \sqrt{x + 5} \sqrt{x + 5}$

Move $- 2$ to the left of $\sqrt{x + 5}$ .

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \cdot \sqrt{x + 5} + \sqrt{x + 5} \sqrt{x + 5}$

Multiply $\sqrt{x + 5} \sqrt{x + 5}$ .

Raise $\sqrt{x + 5}$ to the power of $1$ .

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + {\sqrt{x + 5}}^{1} \sqrt{x + 5}$

Raise $\sqrt{x + 5}$ to the power of $1$ .

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + {\sqrt{x + 5}}^{1} {\sqrt{x + 5}}^{1}$

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

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + {\sqrt{x + 5}}^{1 + 1}$

Add $1$ and $1$ .

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + {\sqrt{x + 5}}^{2}$

Rewrite ${\sqrt{x + 5}}^{2}$ as $x + 5$ .

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

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + {({(x + 5)}^{\frac{1}{2}})}^{2}$

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

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + {(x + 5)}^{\frac{1}{2} \cdot 2}$

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

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + {(x + 5)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + {(x + 5)}^{\frac{2}{2}}$

Rewrite the expression.

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + {(x + 5)}^{1}$

Simplify.

$x + 1 = 4 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + x + 5$

Add $4$ and $5$ .

$x + 1 = 9 - 2 \sqrt{x + 5} - 2 \sqrt{x + 5} + x$

Subtract $2 \sqrt{x + 5}$ from $- 2 \sqrt{x + 5}$ .

$x + 1 = 9 - 4 \sqrt{x + 5} + x$

Step 4

Solve for $- 4 \sqrt{x + 5}$ .

Rewrite the equation as $9 - 4 \sqrt{x + 5} + x = x + 1$ .

$9 - 4 \sqrt{x + 5} + x = x + 1$

Move all terms not containing $- 4 \sqrt{x + 5}$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$- 4 \sqrt{x + 5} + x = x + 1 - 9$

Subtract $x$ from both sides of the equation.

$- 4 \sqrt{x + 5} = x + 1 - 9 - x$

Combine the opposite terms in $x + 1 - 9 - x$ .

Subtract $x$ from $x$ .

$- 4 \sqrt{x + 5} = 0 + 1 - 9$

Add $0$ and $1$ .

$- 4 \sqrt{x + 5} = 1 - 9$

Subtract $9$ from $1$ .

$- 4 \sqrt{x + 5} = - 8$

Step 5

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

${(- 4 \sqrt{x + 5})}^{2} = {(- 8)}^{2}$

Step 6

Simplify each side of the equation.

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

${(- 4 {(x + 5)}^{\frac{1}{2}})}^{2} = {(- 8)}^{2}$

Simplify the left side.

Simplify ${(- 4 {(x + 5)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 4 {(x + 5)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(x + 5)}^{\frac{1}{2}})}^{2} = {(- 8)}^{2}$

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

$16 {({(x + 5)}^{\frac{1}{2}})}^{2} = {(- 8)}^{2}$

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

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

$16 {(x + 5)}^{\frac{1}{2} \cdot 2} = {(- 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x + 5)}^{\frac{1}{2} \cdot 2} = {(- 8)}^{2}$

Rewrite the expression.

$16 {(x + 5)}^{1} = {(- 8)}^{2}$

Simplify.

$16 (x + 5) = {(- 8)}^{2}$

Apply the distributive property.

$16 x + 16 \cdot 5 = {(- 8)}^{2}$

Multiply $16$ by $5$ .

$16 x + 80 = {(- 8)}^{2}$

Simplify the right side.

Raise $- 8$ to the power of $2$ .

$16 x + 80 = 64$

Step 7

Solve for $x$ .

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

Subtract $80$ from both sides of the equation.

$16 x = 64 - 80$

Subtract $80$ from $64$ .

$16 x = - 16$

Divide each term in $16 x = - 16$ by $16$ and simplify.

Divide each term in $16 x = - 16$ by $16$ .

$\frac{16 x}{16} = \frac{- 16}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{- 16}{16}$

Divide $x$ by $1$ .

$x = \frac{- 16}{16}$

Simplify the right side.

Divide $- 16$ by $16$ .

$x = - 1$

Solve for x 2+ square root of x+1- square root of x+5=0

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