Solve by Factoring square root of 13x+1=1+ square root of 11x

$\sqrt{13 x + 1} = 1 + \sqrt{11 x}$

Step 1

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

Subtract $1$ from both sides of the equation.

$\sqrt{13 x + 1} - 1 = \sqrt{11 x}$

Subtract $\sqrt{11 x}$ from both sides of the equation.

$\sqrt{13 x + 1} - 1 - \sqrt{11 x} = 0$

Step 2

Move all terms not containing $\sqrt{13 x + 1}$ to the right side of the equation.

Add $1$ to both sides of the equation.

$\sqrt{13 x + 1} - \sqrt{11 x} = 1$

Add $\sqrt{11 x}$ to both sides of the equation.

$\sqrt{13 x + 1} = 1 + \sqrt{11 x}$

Step 3

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

${\sqrt{13 x + 1}}^{2} = {(1 + \sqrt{11 x})}^{2}$

Step 4

Simplify each side of the equation.

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

${({(13 x + 1)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{11 x})}^{2}$

Simplify the left side.

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

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

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

${(13 x + 1)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{11 x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(13 x + 1)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{11 x})}^{2}$

Rewrite the expression.

${(13 x + 1)}^{1} = {(1 + \sqrt{11 x})}^{2}$

Simplify.

$13 x + 1 = {(1 + \sqrt{11 x})}^{2}$

Simplify the right side.

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

Rewrite ${(1 + \sqrt{11 x})}^{2}$ as $(1 + \sqrt{11 x}) (1 + \sqrt{11 x})$ .

$13 x + 1 = (1 + \sqrt{11 x}) (1 + \sqrt{11 x})$

Expand $(1 + \sqrt{11 x}) (1 + \sqrt{11 x})$ using the FOIL Method.

Apply the distributive property.

$13 x + 1 = 1 (1 + \sqrt{11 x}) + \sqrt{11 x} (1 + \sqrt{11 x})$

Apply the distributive property.

$13 x + 1 = 1 \cdot 1 + 1 \sqrt{11 x} + \sqrt{11 x} (1 + \sqrt{11 x})$

Apply the distributive property.

$13 x + 1 = 1 \cdot 1 + 1 \sqrt{11 x} + \sqrt{11 x} \cdot 1 + \sqrt{11 x} \sqrt{11 x}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$13 x + 1 = 1 + 1 \sqrt{11 x} + \sqrt{11 x} \cdot 1 + \sqrt{11 x} \sqrt{11 x}$

Multiply $\sqrt{11 x}$ by $1$ .

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} \cdot 1 + \sqrt{11 x} \sqrt{11 x}$

Multiply $\sqrt{11 x}$ by $1$ .

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + \sqrt{11 x} \sqrt{11 x}$

Multiply $\sqrt{11 x} \sqrt{11 x}$ .

Raise $\sqrt{11 x}$ to the power of $1$ .

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + {\sqrt{11 x}}^{1} \sqrt{11 x}$

Raise $\sqrt{11 x}$ to the power of $1$ .

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + {\sqrt{11 x}}^{1} {\sqrt{11 x}}^{1}$

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

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + {\sqrt{11 x}}^{1 + 1}$

Add $1$ and $1$ .

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + {\sqrt{11 x}}^{2}$

Rewrite ${\sqrt{11 x}}^{2}$ as $11 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 x}$ as ${(11 x)}^{\frac{1}{2}}$ .

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + {({(11 x)}^{\frac{1}{2}})}^{2}$

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

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + {(11 x)}^{\frac{1}{2} \cdot 2}$

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

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + {(11 x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + {(11 x)}^{\frac{2}{2}}$

Rewrite the expression.

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + {(11 x)}^{1}$

Simplify.

$13 x + 1 = 1 + \sqrt{11 x} + \sqrt{11 x} + 11 x$

Add $\sqrt{11 x}$ and $\sqrt{11 x}$ .

$13 x + 1 = 1 + 2 \sqrt{11 x} + 11 x$

Step 5

Solve for $2 \sqrt{11 x}$ .

Rewrite the equation as $1 + 2 \sqrt{11 x} + 11 x = 13 x + 1$ .

$1 + 2 \sqrt{11 x} + 11 x = 13 x + 1$

Move all terms not containing $2 \sqrt{11 x}$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$2 \sqrt{11 x} + 11 x = 13 x + 1 - 1$

Subtract $11 x$ from both sides of the equation.

$2 \sqrt{11 x} = 13 x + 1 - 1 - 11 x$

Combine the opposite terms in $13 x + 1 - 1 - 11 x$ .

Subtract $1$ from $1$ .

$2 \sqrt{11 x} = 13 x + 0 - 11 x$

Add $13 x$ and $0$ .

$2 \sqrt{11 x} = 13 x - 11 x$

Subtract $11 x$ from $13 x$ .

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

Step 6

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

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

Step 7

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 x}$ as ${(11 x)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${(2 {(11 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $11 x$ .

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $2 \cdot 11^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Apply the product rule to $2 \cdot 11^{\frac{1}{2}}$ .

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

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

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

Multiply the exponents in ${(11^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $4$ by $11$ .

$44 {(x^{\frac{1}{2}})}^{2} = {(2 x)}^{2}$

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

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

$44 x^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$44 x^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Rewrite the expression.

$44 x^{1} = {(2 x)}^{2}$

Simplify.

$44 x = {(2 x)}^{2}$

Simplify the right side.

Simplify ${(2 x)}^{2}$ .

Apply the product rule to $2 x$ .

$44 x = 2^{2} x^{2}$

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

$44 x = 4 x^{2}$

Step 8

Solve for $x$ .

Subtract $4 x^{2}$ from both sides of the equation.

$44 x - 4 x^{2} = 0$

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$44 u - 4 u^{2} = 0$

Factor $4 u$ out of $44 u - 4 u^{2}$ .

Factor $4 u$ out of $44 u$ .

$4 u (11) - 4 u^{2} = 0$

Factor $4 u$ out of $- 4 u^{2}$ .

$4 u (11) + 4 u (- u) = 0$

Factor $4 u$ out of $4 u (11) + 4 u (- u)$ .

$4 u (11 - u) = 0$

Replace all occurrences of $u$ with $x$ .

$4 x (11 - x) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$11 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $11 - x$ equal to $0$ and solve for $x$ .

Set $11 - x$ equal to $0$ .

$11 - x = 0$

Solve $11 - x = 0$ for $x$ .

Subtract $11$ from both sides of the equation.

$- x = - 11$

Divide each term in $- x = - 11$ by $- 1$ and simplify.

Divide each term in $- x = - 11$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 11}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 11}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 11}{- 1}$

Simplify the right side.

Divide $- 11$ by $- 1$ .

$x = 11$

The final solution is all the values that make $4 x (11 - x) = 0$ true.

$x = 0, 11$

Solve by Factoring square root of 13x+1=1+ square root of 11x

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