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

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

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(\sqrt{x - 5} - 1)}^{2}$ .

Rewrite ${(\sqrt{x - 5} - 1)}^{2}$ as $(\sqrt{x - 5} - 1) (\sqrt{x - 5} - 1)$ .

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

Expand $(\sqrt{x - 5} - 1) (\sqrt{x - 5} - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Move $- 1$ to the left of $\sqrt{x - 5}$ .

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

Rewrite $- 1 \sqrt{x - 5}$ as $- \sqrt{x - 5}$ .

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

Rewrite $- 1 \sqrt{x - 5}$ as $- \sqrt{x - 5}$ .

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

Multiply $- 1$ by $- 1$ .

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

Add $- 5$ and $1$ .

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

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

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

Step 4

Solve for $- 2 \sqrt{x - 5}$ .

Rewrite the equation as $x - 4 - 2 \sqrt{x - 5} = x$ .

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

Move all terms not containing $- 2 \sqrt{x - 5}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

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

Add $4$ to both sides of the equation.

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

Subtract $x$ from $x$ .

$- 2 \sqrt{x - 5} = 0 + 4$

Add $0$ and $4$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$4 (x - 5) = 4^{2}$

Apply the distributive property.

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

Multiply $4$ by $- 5$ .

$4 x - 20 = 4^{2}$

Simplify the right side.

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

$4 x - 20 = 16$

Step 7

Solve for $x$ .

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

Add $20$ to both sides of the equation.

$4 x = 16 + 20$

Add $16$ and $20$ .

$4 x = 36$

Divide each term in $4 x = 36$ by $4$ and simplify.

Divide each term in $4 x = 36$ by $4$ .

$\frac{4 x}{4} = \frac{36}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{36}{4}$

Divide $x$ by $1$ .

$x = \frac{36}{4}$

Simplify the right side.

Divide $36$ by $4$ .

$x = 9$

Step 8

Exclude the solutions that do not make $\sqrt{x} + 1 = \sqrt{x - 5}$ true.

$No solution$

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

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