Solve by Factoring square root of x+9=1+ square root of x+2

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $1$ from both sides of the equation.

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

Subtract $\sqrt{x + 2}$ from both sides of the equation.

$\sqrt{x + 9} - 1 - \sqrt{x + 2} = 0$

Step 2

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

Add $1$ to both sides of the equation.

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

Add $\sqrt{x + 2}$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

Expand $(1 + \sqrt{x + 2}) (1 + \sqrt{x + 2})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $\sqrt{x + 2}$ by $1$ .

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

Multiply $\sqrt{x + 2}$ by $1$ .

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

Multiply $\sqrt{x + 2} \sqrt{x + 2}$ .

Raise $\sqrt{x + 2}$ to the power of $1$ .

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

Raise $\sqrt{x + 2}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Add $1$ and $2$ .

$x + 9 = 3 + \sqrt{x + 2} + \sqrt{x + 2} + x$

Add $\sqrt{x + 2}$ and $\sqrt{x + 2}$ .

$x + 9 = 3 + 2 \sqrt{x + 2} + x$

Step 5

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

Rewrite the equation as $3 + 2 \sqrt{x + 2} + x = x + 9$ .

$3 + 2 \sqrt{x + 2} + x = x + 9$

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

Subtract $3$ from both sides of the equation.

$2 \sqrt{x + 2} + x = x + 9 - 3$

Subtract $x$ from both sides of the equation.

$2 \sqrt{x + 2} = x + 9 - 3 - x$

Combine the opposite terms in $x + 9 - 3 - x$ .

Subtract $x$ from $x$ .

$2 \sqrt{x + 2} = 0 + 9 - 3$

Add $0$ and $9$ .

$2 \sqrt{x + 2} = 9 - 3$

Subtract $3$ from $9$ .

$2 \sqrt{x + 2} = 6$

Step 6

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

${(2 \sqrt{x + 2})}^{2} = 6^{2}$

Step 7

Simplify each side of the equation.

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

${(2 {(x + 2)}^{\frac{1}{2}})}^{2} = 6^{2}$

Simplify the left side.

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

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

$2^{2} {({(x + 2)}^{\frac{1}{2}})}^{2} = 6^{2}$

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

$4 {({(x + 2)}^{\frac{1}{2}})}^{2} = 6^{2}$

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

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

$4 {(x + 2)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x + 2)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Rewrite the expression.

$4 {(x + 2)}^{1} = 6^{2}$

Simplify.

$4 (x + 2) = 6^{2}$

Apply the distributive property.

$4 x + 4 \cdot 2 = 6^{2}$

Multiply $4$ by $2$ .

$4 x + 8 = 6^{2}$

Simplify the right side.

Raise $6$ to the power of $2$ .

$4 x + 8 = 36$

Step 8

Solve for $x$ .

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

Subtract $8$ from both sides of the equation.

$4 x = 36 - 8$

Subtract $8$ from $36$ .

$4 x = 28$

Divide each term in $4 x = 28$ by $4$ and simplify.

Divide each term in $4 x = 28$ by $4$ .

$\frac{4 x}{4} = \frac{28}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{28}{4}$

Divide $x$ by $1$ .

$x = \frac{28}{4}$

Simplify the right side.

Divide $28$ by $4$ .

$x = 7$

Solve by Factoring square root of x+9=1+ square root of x+2

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