Solve by Factoring square root of y-5+ square root of y=5

$\sqrt{y - 5} + \sqrt{y} = 5$

Step 1

Subtract $5$ from both sides of the equation.

$\sqrt{y - 5} + \sqrt{y} - 5 = 0$

Step 2

Move all terms not containing $\sqrt{y - 5}$ to the right side of the equation.

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

$\sqrt{y - 5} - 5 = - \sqrt{y}$

Add $5$ to both sides of the equation.

$\sqrt{y - 5} = - \sqrt{y} + 5$

Step 3

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

${\sqrt{y - 5}}^{2} = {(- \sqrt{y} + 5)}^{2}$

Step 4

Simplify each side of the equation.

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

${({(y - 5)}^{\frac{1}{2}})}^{2} = {(- \sqrt{y} + 5)}^{2}$

Simplify the left side.

Simplify ${({(y - 5)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(y - 5)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(y - 5)}^{1} = {(- \sqrt{y} + 5)}^{2}$

Simplify.

$y - 5 = {(- \sqrt{y} + 5)}^{2}$

Simplify the right side.

Simplify ${(- \sqrt{y} + 5)}^{2}$ .

Rewrite ${(- \sqrt{y} + 5)}^{2}$ as $(- \sqrt{y} + 5) (- \sqrt{y} + 5)$ .

$y - 5 = (- \sqrt{y} + 5) (- \sqrt{y} + 5)$

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

Apply the distributive property.

$y - 5 = - \sqrt{y} (- \sqrt{y} + 5) + 5 (- \sqrt{y} + 5)$

Apply the distributive property.

$y - 5 = - \sqrt{y} (- \sqrt{y}) - \sqrt{y} \cdot 5 + 5 (- \sqrt{y} + 5)$

Apply the distributive property.

$y - 5 = - \sqrt{y} (- \sqrt{y}) - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

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

Multiply $- 1$ by $- 1$ .

$y - 5 = 1 \sqrt{y} \sqrt{y} - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

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

$y - 5 = \sqrt{y} \sqrt{y} - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

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

$y - 5 = {\sqrt{y}}^{1} \sqrt{y} - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

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

$y - 5 = {\sqrt{y}}^{1} {\sqrt{y}}^{1} - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

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

$y - 5 = {\sqrt{y}}^{1 + 1} - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

Add $1$ and $1$ .

$y - 5 = {\sqrt{y}}^{2} - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

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

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

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

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

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

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

$y - 5 = y^{\frac{2}{2}} - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

Cancel the common factor of $2$ .

Cancel the common factor.

$y - 5 = y^{\frac{2}{2}} - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

Rewrite the expression.

$y - 5 = y^{1} - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

Simplify.

$y - 5 = y - \sqrt{y} \cdot 5 + 5 (- \sqrt{y}) + 5 \cdot 5$

Multiply $5$ by $- 1$ .

$y - 5 = y - 5 \sqrt{y} + 5 (- \sqrt{y}) + 5 \cdot 5$

Multiply $- 1$ by $5$ .

$y - 5 = y - 5 \sqrt{y} - 5 \sqrt{y} + 5 \cdot 5$

Multiply $5$ by $5$ .

$y - 5 = y - 5 \sqrt{y} - 5 \sqrt{y} + 25$

Subtract $5 \sqrt{y}$ from $- 5 \sqrt{y}$ .

$y - 5 = y - 10 \sqrt{y} + 25$

Step 5

Solve for $- 10 \sqrt{y}$ .

Rewrite the equation as $y - 10 \sqrt{y} + 25 = y - 5$ .

$y - 10 \sqrt{y} + 25 = y - 5$

Move all terms not containing $- 10 \sqrt{y}$ to the right side of the equation.

Subtract $y$ from both sides of the equation.

$- 10 \sqrt{y} + 25 = y - 5 - y$

Subtract $25$ from both sides of the equation.

$- 10 \sqrt{y} = y - 5 - y - 25$

Combine the opposite terms in $y - 5 - y - 25$ .

Subtract $y$ from $y$ .

$- 10 \sqrt{y} = 0 - 5 - 25$

Subtract $5$ from $0$ .

$- 10 \sqrt{y} = - 5 - 25$

Subtract $25$ from $- 5$ .

$- 10 \sqrt{y} = - 30$

Step 6

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

${(- 10 \sqrt{y})}^{2} = {(- 30)}^{2}$

Step 7

Simplify each side of the equation.

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

${(- 10 y^{\frac{1}{2}})}^{2} = {(- 30)}^{2}$

Simplify the left side.

Simplify ${(- 10 y^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 10 y^{\frac{1}{2}}$ .

${(- 10)}^{2} {(y^{\frac{1}{2}})}^{2} = {(- 30)}^{2}$

Raise $- 10$ to the power of $2$ .

$100 {(y^{\frac{1}{2}})}^{2} = {(- 30)}^{2}$

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

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

$100 y^{\frac{1}{2} \cdot 2} = {(- 30)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 y^{\frac{1}{2} \cdot 2} = {(- 30)}^{2}$

Rewrite the expression.

$100 y^{1} = {(- 30)}^{2}$

Simplify.

$100 y = {(- 30)}^{2}$

Simplify the right side.

Raise $- 30$ to the power of $2$ .

$100 y = 900$

Step 8

Divide each term in $100 y = 900$ by $100$ and simplify.

Divide each term in $100 y = 900$ by $100$ .

$\frac{100 y}{100} = \frac{900}{100}$

Simplify the left side.

Cancel the common factor of $100$ .

Cancel the common factor.

$\frac{100 y}{100} = \frac{900}{100}$

Divide $y$ by $1$ .

$y = \frac{900}{100}$

Simplify the right side.

Divide $900$ by $100$ .

$y = 9$

Solve by Factoring square root of y-5+ square root of y=5

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