Solve by Factoring square root of y-11+ square root of y=11

$\sqrt{y - 11} + \sqrt{y} = 11$

Step 1

Subtract $11$ from both sides of the equation.

$\sqrt{y - 11} + \sqrt{y} - 11 = 0$

Step 2

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

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

$\sqrt{y - 11} - 11 = - \sqrt{y}$

Add $11$ to both sides of the equation.

$\sqrt{y - 11} = - \sqrt{y} + 11$

Step 3

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

${\sqrt{y - 11}}^{2} = {(- \sqrt{y} + 11)}^{2}$

Step 4

Simplify each side of the equation.

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

${({(y - 11)}^{\frac{1}{2}})}^{2} = {(- \sqrt{y} + 11)}^{2}$

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$y - 11 = {(- \sqrt{y} + 11)}^{2}$

Simplify the right side.

Simplify ${(- \sqrt{y} + 11)}^{2}$ .

Rewrite ${(- \sqrt{y} + 11)}^{2}$ as $(- \sqrt{y} + 11) (- \sqrt{y} + 11)$ .

$y - 11 = (- \sqrt{y} + 11) (- \sqrt{y} + 11)$

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

Apply the distributive property.

$y - 11 = - \sqrt{y} (- \sqrt{y} + 11) + 11 (- \sqrt{y} + 11)$

Apply the distributive property.

$y - 11 = - \sqrt{y} (- \sqrt{y}) - \sqrt{y} \cdot 11 + 11 (- \sqrt{y} + 11)$

Apply the distributive property.

$y - 11 = - \sqrt{y} (- \sqrt{y}) - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{y} (- \sqrt{y})$ .

Multiply $- 1$ by $- 1$ .

$y - 11 = 1 \sqrt{y} \sqrt{y} - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

Multiply $\sqrt{y}$ by $1$ .

$y - 11 = \sqrt{y} \sqrt{y} - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

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

$y - 11 = {\sqrt{y}}^{1} \sqrt{y} - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

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

$y - 11 = {\sqrt{y}}^{1} {\sqrt{y}}^{1} - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

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

$y - 11 = {\sqrt{y}}^{1 + 1} - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

Add $1$ and $1$ .

$y - 11 = {\sqrt{y}}^{2} - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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

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

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

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

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

$y - 11 = y^{\frac{2}{2}} - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

Cancel the common factor of $2$ .

Cancel the common factor.

$y - 11 = y^{\frac{2}{2}} - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

Rewrite the expression.

$y - 11 = y^{1} - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

Simplify.

$y - 11 = y - \sqrt{y} \cdot 11 + 11 (- \sqrt{y}) + 11 \cdot 11$

Multiply $11$ by $- 1$ .

$y - 11 = y - 11 \sqrt{y} + 11 (- \sqrt{y}) + 11 \cdot 11$

Multiply $- 1$ by $11$ .

$y - 11 = y - 11 \sqrt{y} - 11 \sqrt{y} + 11 \cdot 11$

Multiply $11$ by $11$ .

$y - 11 = y - 11 \sqrt{y} - 11 \sqrt{y} + 121$

Subtract $11 \sqrt{y}$ from $- 11 \sqrt{y}$ .

$y - 11 = y - 22 \sqrt{y} + 121$

Step 5

Solve for $- 22 \sqrt{y}$ .

Rewrite the equation as $y - 22 \sqrt{y} + 121 = y - 11$ .

$y - 22 \sqrt{y} + 121 = y - 11$

Move all terms not containing $- 22 \sqrt{y}$ to the right side of the equation.

Subtract $y$ from both sides of the equation.

$- 22 \sqrt{y} + 121 = y - 11 - y$

Subtract $121$ from both sides of the equation.

$- 22 \sqrt{y} = y - 11 - y - 121$

Combine the opposite terms in $y - 11 - y - 121$ .

Subtract $y$ from $y$ .

$- 22 \sqrt{y} = 0 - 11 - 121$

Subtract $11$ from $0$ .

$- 22 \sqrt{y} = - 11 - 121$

Subtract $121$ from $- 11$ .

$- 22 \sqrt{y} = - 132$

Step 6

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

${(- 22 \sqrt{y})}^{2} = {(- 132)}^{2}$

Step 7

Simplify each side of the equation.

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

${(- 22 y^{\frac{1}{2}})}^{2} = {(- 132)}^{2}$

Simplify the left side.

Simplify ${(- 22 y^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 22 y^{\frac{1}{2}}$ .

${(- 22)}^{2} {(y^{\frac{1}{2}})}^{2} = {(- 132)}^{2}$

Raise $- 22$ to the power of $2$ .

$484 {(y^{\frac{1}{2}})}^{2} = {(- 132)}^{2}$

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

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

$484 y^{\frac{1}{2} \cdot 2} = {(- 132)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$484 y^{\frac{1}{2} \cdot 2} = {(- 132)}^{2}$

Rewrite the expression.

$484 y^{1} = {(- 132)}^{2}$

Simplify.

$484 y = {(- 132)}^{2}$

Simplify the right side.

Raise $- 132$ to the power of $2$ .

$484 y = 17424$

Step 8

Divide each term in $484 y = 17424$ by $484$ and simplify.

Divide each term in $484 y = 17424$ by $484$ .

$\frac{484 y}{484} = \frac{17424}{484}$

Simplify the left side.

Cancel the common factor of $484$ .

Cancel the common factor.

$\frac{484 y}{484} = \frac{17424}{484}$

Divide $y$ by $1$ .

$y = \frac{17424}{484}$

Simplify the right side.

Divide $17424$ by $484$ .

$y = 36$

Solve by Factoring square root of y-11+ square root of y=11

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