Solve the Rational Equation for x square root of x+ square root of x-5=1

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

Step 1

Subtract $\sqrt{x - 5}$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $- \sqrt{x - 5}$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{x - 5}$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Subtract $5$ from $1$ .

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

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

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

Step 4

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

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

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

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

Add $4$ to both sides of the equation.

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

Subtract $x$ from both sides of the equation.

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

Combine the opposite terms in $x + 4 - x$ .

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

$No solution$

Solve the Rational Equation for x square root of x+ square root of x-5=1

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