Solve the Rational Equation for x square root of x+1 = square root of x+9

$\sqrt{x} + 1 = \sqrt{x + 9}$

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

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

${\sqrt{x}}^{2} = {(\sqrt{x + 9} - 1)}^{2}$

Step 3

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = {(\sqrt{x + 9} - 1)}^{2}$

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

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 9} - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 9} - 1)}^{2}$

Rewrite the expression.

$x^{1} = {(\sqrt{x + 9} - 1)}^{2}$

Simplify.

$x = {(\sqrt{x + 9} - 1)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x + 9} - 1)}^{2}$ .

Rewrite ${(\sqrt{x + 9} - 1)}^{2}$ as $(\sqrt{x + 9} - 1) (\sqrt{x + 9} - 1)$ .

$x = (\sqrt{x + 9} - 1) (\sqrt{x + 9} - 1)$

Expand $(\sqrt{x + 9} - 1) (\sqrt{x + 9} - 1)$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{x + 9} (\sqrt{x + 9} - 1) - 1 (\sqrt{x + 9} - 1)$

Apply the distributive property.

$x = \sqrt{x + 9} \sqrt{x + 9} + \sqrt{x + 9} \cdot - 1 - 1 (\sqrt{x + 9} - 1)$

Apply the distributive property.

$x = \sqrt{x + 9} \sqrt{x + 9} + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 9} \sqrt{x + 9}$ .

Raise $\sqrt{x + 9}$ to the power of $1$ .

$x = {\sqrt{x + 9}}^{1} \sqrt{x + 9} + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

Raise $\sqrt{x + 9}$ to the power of $1$ .

$x = {\sqrt{x + 9}}^{1} {\sqrt{x + 9}}^{1} + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

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

$x = {\sqrt{x + 9}}^{1 + 1} + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x + 9}}^{2}$ as $x + 9$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 9}$ as ${(x + 9)}^{\frac{1}{2}}$ .

$x = {({(x + 9)}^{\frac{1}{2}})}^{2} + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

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

$x = {(x + 9)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

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

$x = {(x + 9)}^{\frac{2}{2}} + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = {(x + 9)}^{\frac{2}{2}} + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

Rewrite the expression.

$x = {(x + 9)}^{1} + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

Simplify.

$x = x + 9 + \sqrt{x + 9} \cdot - 1 - 1 \sqrt{x + 9} - 1 \cdot - 1$

Move $- 1$ to the left of $\sqrt{x + 9}$ .

$x = x + 9 - 1 \cdot \sqrt{x + 9} - 1 \sqrt{x + 9} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{x + 9}$ as $- \sqrt{x + 9}$ .

$x = x + 9 - \sqrt{x + 9} - 1 \sqrt{x + 9} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{x + 9}$ as $- \sqrt{x + 9}$ .

$x = x + 9 - \sqrt{x + 9} - \sqrt{x + 9} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

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

Add $9$ and $1$ .

$x = x + 10 - \sqrt{x + 9} - \sqrt{x + 9}$

Subtract $\sqrt{x + 9}$ from $- \sqrt{x + 9}$ .

$x = x + 10 - 2 \sqrt{x + 9}$

Step 4

Solve for $- 2 \sqrt{x + 9}$ .

Rewrite the equation as $x + 10 - 2 \sqrt{x + 9} = x$ .

$x + 10 - 2 \sqrt{x + 9} = x$

Move all terms not containing $- 2 \sqrt{x + 9}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$10 - 2 \sqrt{x + 9} = x - x$

Subtract $10$ from both sides of the equation.

$- 2 \sqrt{x + 9} = x - x - 10$

Subtract $x$ from $x$ .

$- 2 \sqrt{x + 9} = 0 - 10$

Subtract $10$ from $0$ .

$- 2 \sqrt{x + 9} = - 10$

Step 5

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

${(- 2 \sqrt{x + 9})}^{2} = {(- 10)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 9}$ as ${(x + 9)}^{\frac{1}{2}}$ .

${(- 2 {(x + 9)}^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

Simplify the left side.

Simplify ${(- 2 {(x + 9)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 2 {(x + 9)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(x + 9)}^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

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

$4 {({(x + 9)}^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

Multiply the exponents in ${({(x + 9)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(x + 9)}^{\frac{1}{2} \cdot 2} = {(- 10)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x + 9)}^{\frac{1}{2} \cdot 2} = {(- 10)}^{2}$

Rewrite the expression.

$4 {(x + 9)}^{1} = {(- 10)}^{2}$

Simplify.

$4 (x + 9) = {(- 10)}^{2}$

Apply the distributive property.

$4 x + 4 \cdot 9 = {(- 10)}^{2}$

Multiply $4$ by $9$ .

$4 x + 36 = {(- 10)}^{2}$

Simplify the right side.

Raise $- 10$ to the power of $2$ .

$4 x + 36 = 100$

Step 7

Solve for $x$ .

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

Subtract $36$ from both sides of the equation.

$4 x = 100 - 36$

Subtract $36$ from $100$ .

$4 x = 64$

Divide each term in $4 x = 64$ by $4$ and simplify.

Divide each term in $4 x = 64$ by $4$ .

$\frac{4 x}{4} = \frac{64}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{64}{4}$

Divide $x$ by $1$ .

$x = \frac{64}{4}$

Simplify the right side.

Divide $64$ by $4$ .

$x = 16$

Solve the Rational Equation for x square root of x+1 = square root of x+9

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