Solve for x square root of 9x+1 = square root of 5x+11

$\sqrt{9 x} + 1 = \sqrt{5 x + 11}$

Subtract $1$ from both sides of the equation.

$\sqrt{9 x} = \sqrt{5 x + 11} - 1$

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

${\sqrt{9 x}}^{2} = {(\sqrt{5 x + 11} - 1)}^{2}$

Simplify each side of the equation.

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

${({(9 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{5 x + 11} - 1)}^{2}$

Simplify the left side.

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

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

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

${(9 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{5 x + 11} - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(9 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{5 x + 11} - 1)}^{2}$

Rewrite the expression.

${(9 x)}^{1} = {(\sqrt{5 x + 11} - 1)}^{2}$

Simplify.

$9 x = {(\sqrt{5 x + 11} - 1)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{5 x + 11} - 1)}^{2}$ .

Rewrite ${(\sqrt{5 x + 11} - 1)}^{2}$ as $(\sqrt{5 x + 11} - 1) (\sqrt{5 x + 11} - 1)$ .

$9 x = (\sqrt{5 x + 11} - 1) (\sqrt{5 x + 11} - 1)$

Expand $(\sqrt{5 x + 11} - 1) (\sqrt{5 x + 11} - 1)$ using the FOIL Method.

Apply the distributive property.

$9 x = \sqrt{5 x + 11} (\sqrt{5 x + 11} - 1) - 1 (\sqrt{5 x + 11} - 1)$

Apply the distributive property.

$9 x = \sqrt{5 x + 11} \sqrt{5 x + 11} + \sqrt{5 x + 11} \cdot - 1 - 1 (\sqrt{5 x + 11} - 1)$

Apply the distributive property.

$9 x = \sqrt{5 x + 11} \sqrt{5 x + 11} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{5 x + 11} \sqrt{5 x + 11}$ .

Raise $\sqrt{5 x + 11}$ to the power of $1$ .

$9 x = {\sqrt{5 x + 11}}^{1} \sqrt{5 x + 11} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Raise $\sqrt{5 x + 11}$ to the power of $1$ .

$9 x = {\sqrt{5 x + 11}}^{1} {\sqrt{5 x + 11}}^{1} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

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

$9 x = {\sqrt{5 x + 11}}^{1 + 1} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Add $1$ and $1$ .

$9 x = {\sqrt{5 x + 11}}^{2} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Rewrite ${\sqrt{5 x + 11}}^{2}$ as $5 x + 11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 11}$ as ${(5 x + 11)}^{\frac{1}{2}}$ .

$9 x = {({(5 x + 11)}^{\frac{1}{2}})}^{2} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

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

$9 x = {(5 x + 11)}^{\frac{1}{2} \cdot 2} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

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

$9 x = {(5 x + 11)}^{\frac{2}{2}} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 x = {(5 x + 11)}^{\frac{2}{2}} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Rewrite the expression.

$9 x = {(5 x + 11)}^{1} + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Simplify.

$9 x = 5 x + 11 + \sqrt{5 x + 11} \cdot - 1 - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Move $- 1$ to the left of $\sqrt{5 x + 11}$ .

$9 x = 5 x + 11 - 1 \cdot \sqrt{5 x + 11} - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{5 x + 11}$ as $- \sqrt{5 x + 11}$ .

$9 x = 5 x + 11 - \sqrt{5 x + 11} - 1 \sqrt{5 x + 11} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{5 x + 11}$ as $- \sqrt{5 x + 11}$ .

$9 x = 5 x + 11 - \sqrt{5 x + 11} - \sqrt{5 x + 11} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$9 x = 5 x + 11 - \sqrt{5 x + 11} - \sqrt{5 x + 11} + 1$

Add $11$ and $1$ .

$9 x = 5 x + 12 - \sqrt{5 x + 11} - \sqrt{5 x + 11}$

Subtract $\sqrt{5 x + 11}$ from $- \sqrt{5 x + 11}$ .

$9 x = 5 x + 12 - 2 \sqrt{5 x + 11}$

Solve for $- 2 \sqrt{5 x + 11}$ .

Rewrite the equation as $5 x + 12 - 2 \sqrt{5 x + 11} = 9 x$ .

$5 x + 12 - 2 \sqrt{5 x + 11} = 9 x$

Move all terms not containing $- 2 \sqrt{5 x + 11}$ to the right side of the equation.

Subtract $5 x$ from both sides of the equation.

$12 - 2 \sqrt{5 x + 11} = 9 x - 5 x$

Subtract $12$ from both sides of the equation.

$- 2 \sqrt{5 x + 11} = 9 x - 5 x - 12$

Subtract $5 x$ from $9 x$ .

$- 2 \sqrt{5 x + 11} = 4 x - 12$

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

${(- 2 \sqrt{5 x + 11})}^{2} = {(4 x - 12)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 11}$ as ${(5 x + 11)}^{\frac{1}{2}}$ .

${(- 2 {(5 x + 11)}^{\frac{1}{2}})}^{2} = {(4 x - 12)}^{2}$

Simplify the left side.

Simplify ${(- 2 {(5 x + 11)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 2 {(5 x + 11)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(5 x + 11)}^{\frac{1}{2}})}^{2} = {(4 x - 12)}^{2}$

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

$4 {({(5 x + 11)}^{\frac{1}{2}})}^{2} = {(4 x - 12)}^{2}$

Multiply the exponents in ${({(5 x + 11)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(5 x + 11)}^{\frac{1}{2} \cdot 2} = {(4 x - 12)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(5 x + 11)}^{\frac{1}{2} \cdot 2} = {(4 x - 12)}^{2}$

Rewrite the expression.

$4 {(5 x + 11)}^{1} = {(4 x - 12)}^{2}$

Simplify.

$4 (5 x + 11) = {(4 x - 12)}^{2}$

Apply the distributive property.

$4 (5 x) + 4 \cdot 11 = {(4 x - 12)}^{2}$

Multiply.

Multiply $5$ by $4$ .

$20 x + 4 \cdot 11 = {(4 x - 12)}^{2}$

Multiply $4$ by $11$ .

$20 x + 44 = {(4 x - 12)}^{2}$

Simplify the right side.

Simplify ${(4 x - 12)}^{2}$ .

Rewrite ${(4 x - 12)}^{2}$ as $(4 x - 12) (4 x - 12)$ .

$20 x + 44 = (4 x - 12) (4 x - 12)$

Expand $(4 x - 12) (4 x - 12)$ using the FOIL Method.

Apply the distributive property.

$20 x + 44 = 4 x (4 x - 12) - 12 (4 x - 12)$

Apply the distributive property.

$20 x + 44 = 4 x (4 x) + 4 x \cdot - 12 - 12 (4 x - 12)$

Apply the distributive property.

$20 x + 44 = 4 x (4 x) + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$20 x + 44 = 4 \cdot 4 x \cdot x + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$20 x + 44 = 4 \cdot 4 (x \cdot x) + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12$

Multiply $x$ by $x$ .

$20 x + 44 = 4 \cdot 4 x^{2} + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12$

Multiply $4$ by $4$ .

$20 x + 44 = 16 x^{2} + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12$

Multiply $- 12$ by $4$ .

$20 x + 44 = 16 x^{2} - 48 x - 12 (4 x) - 12 \cdot - 12$

Multiply $4$ by $- 12$ .

$20 x + 44 = 16 x^{2} - 48 x - 48 x - 12 \cdot - 12$

Multiply $- 12$ by $- 12$ .

$20 x + 44 = 16 x^{2} - 48 x - 48 x + 144$

Subtract $48 x$ from $- 48 x$ .

$20 x + 44 = 16 x^{2} - 96 x + 144$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$16 x^{2} - 96 x + 144 = 20 x + 44$

Move all terms containing $x$ to the left side of the equation.

Subtract $20 x$ from both sides of the equation.

$16 x^{2} - 96 x + 144 - 20 x = 44$

Subtract $20 x$ from $- 96 x$ .

$16 x^{2} - 116 x + 144 = 44$

Subtract $44$ from both sides of the equation.

$16 x^{2} - 116 x + 144 - 44 = 0$

Subtract $44$ from $144$ .

$16 x^{2} - 116 x + 100 = 0$

Factor the left side of the equation.

Factor $4$ out of $16 x^{2} - 116 x + 100$ .

Factor $4$ out of $16 x^{2}$ .

$4 (4 x^{2}) - 116 x + 100 = 0$

Factor $4$ out of $- 116 x$ .

$4 (4 x^{2}) + 4 (- 29 x) + 100 = 0$

Factor $4$ out of $100$ .

$4 (4 x^{2}) + 4 (- 29 x) + 4 (25) = 0$

Factor $4$ out of $4 (4 x^{2}) + 4 (- 29 x)$ .

$4 (4 x^{2} - 29 x) + 4 (25) = 0$

Factor $4$ out of $4 (4 x^{2} - 29 x) + 4 (25)$ .

$4 (4 x^{2} - 29 x + 25) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 25 = 100$ and whose sum is $b = - 29$ .

Factor $- 29$ out of $- 29 x$ .

$4 (4 x^{2} - 29 x + 25) = 0$

Rewrite $- 29$ as $- 4$ plus $- 25$

$4 (4 x^{2} + (- 4 - 25) x + 25) = 0$

Apply the distributive property.

$4 (4 x^{2} - 4 x - 25 x + 25) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((4 x^{2} - 4 x) - 25 x + 25) = 0$

Factor out the greatest common factor (GCF) from each group.

$4 (4 x (x - 1) - 25 (x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$4 ((x - 1) (4 x - 25)) = 0$

Remove unnecessary parentheses.

$4 (x - 1) (4 x - 25) = 0$

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

$x - 1 = 0$

$4 x - 25 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $4 x - 25$ equal to $0$ and solve for $x$ .

Set $4 x - 25$ equal to $0$ .

$4 x - 25 = 0$

Solve $4 x - 25 = 0$ for $x$ .

Add $25$ to both sides of the equation.

$4 x = 25$

Divide each term in $4 x = 25$ by $4$ and simplify.

Divide each term in $4 x = 25$ by $4$ .

$\frac{4 x}{4} = \frac{25}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{25}{4}$

Divide $x$ by $1$ .

$x = \frac{25}{4}$

The final solution is all the values that make $4 (x - 1) (4 x - 25) = 0$ true.

$x = 1, \frac{25}{4}$

Exclude the solutions that do not make $\sqrt{9 x} + 1 = \sqrt{5 x + 11}$ true.

$x = 1$

Solve for x square root of 9x+1 = square root of 5x+11

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