Solve the Rational Equation for x square root of x+2 = square root of x+9

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

Subtract $2$ from both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

Expand $(\sqrt{x + 9} - 2) (\sqrt{x + 9} - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Move $- 2$ to the left of $\sqrt{x + 9}$ .

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

Multiply $- 2$ by $- 2$ .

$x = x + 9 - 2 \sqrt{x + 9} - 2 \sqrt{x + 9} + 4$

Add $9$ and $4$ .

$x = x + 13 - 2 \sqrt{x + 9} - 2 \sqrt{x + 9}$

Subtract $2 \sqrt{x + 9}$ from $- 2 \sqrt{x + 9}$ .

$x = x + 13 - 4 \sqrt{x + 9}$

Solve for $- 4 \sqrt{x + 9}$ .

Rewrite the equation as $x + 13 - 4 \sqrt{x + 9} = x$ .

$x + 13 - 4 \sqrt{x + 9} = x$

Move all terms not containing $- 4 \sqrt{x + 9}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$13 - 4 \sqrt{x + 9} = x - x$

Subtract $13$ from both sides of the equation.

$- 4 \sqrt{x + 9} = x - x - 13$

Subtract $x$ from $x$ .

$- 4 \sqrt{x + 9} = 0 - 13$

Subtract $13$ from $0$ .

$- 4 \sqrt{x + 9} = - 13$

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

${(- 4 \sqrt{x + 9})}^{2} = {(- 13)}^{2}$

Simplify each side of the equation.

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

${(- 4 {(x + 9)}^{\frac{1}{2}})}^{2} = {(- 13)}^{2}$

Simplify the left side.

Simplify ${(- 4 {(x + 9)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 4 {(x + 9)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(x + 9)}^{\frac{1}{2}})}^{2} = {(- 13)}^{2}$

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

$16 {({(x + 9)}^{\frac{1}{2}})}^{2} = {(- 13)}^{2}$

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

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

$16 {(x + 9)}^{\frac{1}{2} \cdot 2} = {(- 13)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x + 9)}^{\frac{1}{2} \cdot 2} = {(- 13)}^{2}$

Rewrite the expression.

$16 {(x + 9)}^{1} = {(- 13)}^{2}$

Simplify.

$16 (x + 9) = {(- 13)}^{2}$

Apply the distributive property.

$16 x + 16 \cdot 9 = {(- 13)}^{2}$

Multiply $16$ by $9$ .

$16 x + 144 = {(- 13)}^{2}$

Simplify the right side.

Raise $- 13$ to the power of $2$ .

$16 x + 144 = 169$

Solve for $x$ .

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

Subtract $144$ from both sides of the equation.

$16 x = 169 - 144$

Subtract $144$ from $169$ .

$16 x = 25$

Divide each term in $16 x = 25$ by $16$ and simplify.

Divide each term in $16 x = 25$ by $16$ .

$\frac{16 x}{16} = \frac{25}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{25}{16}$

Divide $x$ by $1$ .

$x = \frac{25}{16}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{25}{16}$

Decimal Form:

$x = 1.5625$

Mixed Number Form:

$x = 1 \frac{9}{16}$

Solve the Rational Equation for x square root of x+2 = square root of x+9

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