Solve by Factoring square root of x+13=1+ square root of x+1

$\sqrt{x + 13} = 1 + \sqrt{x + 1}$

Step 1

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

Subtract $1$ from both sides of the equation.

$\sqrt{x + 13} - 1 = \sqrt{x + 1}$

Subtract $\sqrt{x + 1}$ from both sides of the equation.

$\sqrt{x + 13} - 1 - \sqrt{x + 1} = 0$

Step 2

Move all terms not containing $\sqrt{x + 13}$ to the right side of the equation.

Add $1$ to both sides of the equation.

$\sqrt{x + 13} - \sqrt{x + 1} = 1$

Add $\sqrt{x + 1}$ to both sides of the equation.

$\sqrt{x + 13} = 1 + \sqrt{x + 1}$

Step 3

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

${\sqrt{x + 13}}^{2} = {(1 + \sqrt{x + 1})}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 13}$ as ${(x + 13)}^{\frac{1}{2}}$ .

${({(x + 13)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{x + 1})}^{2}$

Simplify the left side.

Simplify ${({(x + 13)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 13)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 13)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x + 1})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 13)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x + 1})}^{2}$

Rewrite the expression.

${(x + 13)}^{1} = {(1 + \sqrt{x + 1})}^{2}$

Simplify.

$x + 13 = {(1 + \sqrt{x + 1})}^{2}$

Simplify the right side.

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

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

$x + 13 = (1 + \sqrt{x + 1}) (1 + \sqrt{x + 1})$

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

Apply the distributive property.

$x + 13 = 1 (1 + \sqrt{x + 1}) + \sqrt{x + 1} (1 + \sqrt{x + 1})$

Apply the distributive property.

$x + 13 = 1 \cdot 1 + 1 \sqrt{x + 1} + \sqrt{x + 1} (1 + \sqrt{x + 1})$

Apply the distributive property.

$x + 13 = 1 \cdot 1 + 1 \sqrt{x + 1} + \sqrt{x + 1} \cdot 1 + \sqrt{x + 1} \sqrt{x + 1}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$x + 13 = 1 + 1 \sqrt{x + 1} + \sqrt{x + 1} \cdot 1 + \sqrt{x + 1} \sqrt{x + 1}$

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

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} \cdot 1 + \sqrt{x + 1} \sqrt{x + 1}$

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

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + \sqrt{x + 1} \sqrt{x + 1}$

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

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

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + {\sqrt{x + 1}}^{1} \sqrt{x + 1}$

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

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + {\sqrt{x + 1}}^{1} {\sqrt{x + 1}}^{1}$

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

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + {\sqrt{x + 1}}^{1 + 1}$

Add $1$ and $1$ .

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + {\sqrt{x + 1}}^{2}$

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

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

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + {({(x + 1)}^{\frac{1}{2}})}^{2}$

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

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + {(x + 1)}^{\frac{1}{2} \cdot 2}$

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

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + {(x + 1)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + {(x + 1)}^{\frac{2}{2}}$

Rewrite the expression.

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + {(x + 1)}^{1}$

Simplify.

$x + 13 = 1 + \sqrt{x + 1} + \sqrt{x + 1} + x + 1$

Add $1$ and $1$ .

$x + 13 = 2 + \sqrt{x + 1} + \sqrt{x + 1} + x$

Add $\sqrt{x + 1}$ and $\sqrt{x + 1}$ .

$x + 13 = 2 + 2 \sqrt{x + 1} + x$

Step 5

Solve for $2 \sqrt{x + 1}$ .

Rewrite the equation as $2 + 2 \sqrt{x + 1} + x = x + 13$ .

$2 + 2 \sqrt{x + 1} + x = x + 13$

Move all terms not containing $2 \sqrt{x + 1}$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$2 \sqrt{x + 1} + x = x + 13 - 2$

Subtract $x$ from both sides of the equation.

$2 \sqrt{x + 1} = x + 13 - 2 - x$

Combine the opposite terms in $x + 13 - 2 - x$ .

Subtract $x$ from $x$ .

$2 \sqrt{x + 1} = 0 + 13 - 2$

Add $0$ and $13$ .

$2 \sqrt{x + 1} = 13 - 2$

Subtract $2$ from $13$ .

$2 \sqrt{x + 1} = 11$

Step 6

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

${(2 \sqrt{x + 1})}^{2} = 11^{2}$

Step 7

Simplify each side of the equation.

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

${(2 {(x + 1)}^{\frac{1}{2}})}^{2} = 11^{2}$

Simplify the left side.

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

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

$2^{2} {({(x + 1)}^{\frac{1}{2}})}^{2} = 11^{2}$

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

$4 {({(x + 1)}^{\frac{1}{2}})}^{2} = 11^{2}$

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

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

$4 {(x + 1)}^{\frac{1}{2} \cdot 2} = 11^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x + 1)}^{\frac{1}{2} \cdot 2} = 11^{2}$

Rewrite the expression.

$4 {(x + 1)}^{1} = 11^{2}$

Simplify.

$4 (x + 1) = 11^{2}$

Apply the distributive property.

$4 x + 4 \cdot 1 = 11^{2}$

Multiply $4$ by $1$ .

$4 x + 4 = 11^{2}$

Simplify the right side.

Raise $11$ to the power of $2$ .

$4 x + 4 = 121$

Step 8

Solve for $x$ .

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

Subtract $4$ from both sides of the equation.

$4 x = 121 - 4$

Subtract $4$ from $121$ .

$4 x = 117$

Divide each term in $4 x = 117$ by $4$ and simplify.

Divide each term in $4 x = 117$ by $4$ .

$\frac{4 x}{4} = \frac{117}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{117}{4}$

Divide $x$ by $1$ .

$x = \frac{117}{4}$

Solve by Factoring square root of x+13=1+ square root of x+1

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