Solve for y square root of 4y+16- square root of y-5=6

$\sqrt{4 y + 16} - \sqrt{y - 5} = 6$

Step 1

Add $\sqrt{y - 5}$ to both sides of the equation.

$\sqrt{4 y + 16} = 6 + \sqrt{y - 5}$

Step 2

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

${\sqrt{4 y + 16}}^{2} = {(6 + \sqrt{y - 5})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 y + 16}$ as ${(4 y + 16)}^{\frac{1}{2}}$ .

${({(4 y + 16)}^{\frac{1}{2}})}^{2} = {(6 + \sqrt{y - 5})}^{2}$

Simplify the left side.

Simplify ${({(4 y + 16)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 y + 16)}^{\frac{1}{2}})}^{2}$ .

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

${(4 y + 16)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{y - 5})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 y + 16)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{y - 5})}^{2}$

Rewrite the expression.

${(4 y + 16)}^{1} = {(6 + \sqrt{y - 5})}^{2}$

Simplify.

$4 y + 16 = {(6 + \sqrt{y - 5})}^{2}$

Simplify the right side.

Simplify ${(6 + \sqrt{y - 5})}^{2}$ .

Rewrite ${(6 + \sqrt{y - 5})}^{2}$ as $(6 + \sqrt{y - 5}) (6 + \sqrt{y - 5})$ .

$4 y + 16 = (6 + \sqrt{y - 5}) (6 + \sqrt{y - 5})$

Expand $(6 + \sqrt{y - 5}) (6 + \sqrt{y - 5})$ using the FOIL Method.

Apply the distributive property.

$4 y + 16 = 6 (6 + \sqrt{y - 5}) + \sqrt{y - 5} (6 + \sqrt{y - 5})$

Apply the distributive property.

$4 y + 16 = 6 \cdot 6 + 6 \sqrt{y - 5} + \sqrt{y - 5} (6 + \sqrt{y - 5})$

Apply the distributive property.

$4 y + 16 = 6 \cdot 6 + 6 \sqrt{y - 5} + \sqrt{y - 5} \cdot 6 + \sqrt{y - 5} \sqrt{y - 5}$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$4 y + 16 = 36 + 6 \sqrt{y - 5} + \sqrt{y - 5} \cdot 6 + \sqrt{y - 5} \sqrt{y - 5}$

Move $6$ to the left of $\sqrt{y - 5}$ .

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \cdot \sqrt{y - 5} + \sqrt{y - 5} \sqrt{y - 5}$

Multiply $\sqrt{y - 5} \sqrt{y - 5}$ .

Raise $\sqrt{y - 5}$ to the power of $1$ .

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + {\sqrt{y - 5}}^{1} \sqrt{y - 5}$

Raise $\sqrt{y - 5}$ to the power of $1$ .

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + {\sqrt{y - 5}}^{1} {\sqrt{y - 5}}^{1}$

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

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + {\sqrt{y - 5}}^{1 + 1}$

Add $1$ and $1$ .

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + {\sqrt{y - 5}}^{2}$

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

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

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + {({(y - 5)}^{\frac{1}{2}})}^{2}$

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

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + {(y - 5)}^{\frac{1}{2} \cdot 2}$

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

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + {(y - 5)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + {(y - 5)}^{\frac{2}{2}}$

Rewrite the expression.

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + {(y - 5)}^{1}$

Simplify.

$4 y + 16 = 36 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + y - 5$

Subtract $5$ from $36$ .

$4 y + 16 = 31 + 6 \sqrt{y - 5} + 6 \sqrt{y - 5} + y$

Add $6 \sqrt{y - 5}$ and $6 \sqrt{y - 5}$ .

$4 y + 16 = 31 + 12 \sqrt{y - 5} + y$

Step 4

Solve for $12 \sqrt{y - 5}$ .

Rewrite the equation as $31 + 12 \sqrt{y - 5} + y = 4 y + 16$ .

$31 + 12 \sqrt{y - 5} + y = 4 y + 16$

Move all terms not containing $12 \sqrt{y - 5}$ to the right side of the equation.

Subtract $31$ from both sides of the equation.

$12 \sqrt{y - 5} + y = 4 y + 16 - 31$

Subtract $y$ from both sides of the equation.

$12 \sqrt{y - 5} = 4 y + 16 - 31 - y$

Subtract $y$ from $4 y$ .

$12 \sqrt{y - 5} = 3 y + 16 - 31$

Subtract $31$ from $16$ .

$12 \sqrt{y - 5} = 3 y - 15$

Step 5

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

${(12 \sqrt{y - 5})}^{2} = {(3 y - 15)}^{2}$

Step 6

Simplify each side of the equation.

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

${(12 {(y - 5)}^{\frac{1}{2}})}^{2} = {(3 y - 15)}^{2}$

Simplify the left side.

Simplify ${(12 {(y - 5)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $12 {(y - 5)}^{\frac{1}{2}}$ .

$12^{2} {({(y - 5)}^{\frac{1}{2}})}^{2} = {(3 y - 15)}^{2}$

Raise $12$ to the power of $2$ .

$144 {({(y - 5)}^{\frac{1}{2}})}^{2} = {(3 y - 15)}^{2}$

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

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

$144 {(y - 5)}^{\frac{1}{2} \cdot 2} = {(3 y - 15)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$144 {(y - 5)}^{\frac{1}{2} \cdot 2} = {(3 y - 15)}^{2}$

Rewrite the expression.

$144 {(y - 5)}^{1} = {(3 y - 15)}^{2}$

Simplify.

$144 (y - 5) = {(3 y - 15)}^{2}$

Apply the distributive property.

$144 y + 144 \cdot - 5 = {(3 y - 15)}^{2}$

Multiply $144$ by $- 5$ .

$144 y - 720 = {(3 y - 15)}^{2}$

Simplify the right side.

Simplify ${(3 y - 15)}^{2}$ .

Rewrite ${(3 y - 15)}^{2}$ as $(3 y - 15) (3 y - 15)$ .

$144 y - 720 = (3 y - 15) (3 y - 15)$

Expand $(3 y - 15) (3 y - 15)$ using the FOIL Method.

Apply the distributive property.

$144 y - 720 = 3 y (3 y - 15) - 15 (3 y - 15)$

Apply the distributive property.

$144 y - 720 = 3 y (3 y) + 3 y \cdot - 15 - 15 (3 y - 15)$

Apply the distributive property.

$144 y - 720 = 3 y (3 y) + 3 y \cdot - 15 - 15 (3 y) - 15 \cdot - 15$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$144 y - 720 = 3 \cdot 3 y \cdot y + 3 y \cdot - 15 - 15 (3 y) - 15 \cdot - 15$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$144 y - 720 = 3 \cdot 3 (y \cdot y) + 3 y \cdot - 15 - 15 (3 y) - 15 \cdot - 15$

Multiply $y$ by $y$ .

$144 y - 720 = 3 \cdot 3 y^{2} + 3 y \cdot - 15 - 15 (3 y) - 15 \cdot - 15$

Multiply $3$ by $3$ .

$144 y - 720 = 9 y^{2} + 3 y \cdot - 15 - 15 (3 y) - 15 \cdot - 15$

Multiply $- 15$ by $3$ .

$144 y - 720 = 9 y^{2} - 45 y - 15 (3 y) - 15 \cdot - 15$

Multiply $3$ by $- 15$ .

$144 y - 720 = 9 y^{2} - 45 y - 45 y - 15 \cdot - 15$

Multiply $- 15$ by $- 15$ .

$144 y - 720 = 9 y^{2} - 45 y - 45 y + 225$

Subtract $45 y$ from $- 45 y$ .

$144 y - 720 = 9 y^{2} - 90 y + 225$

Step 7

Solve for $y$ .

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

$9 y^{2} - 90 y + 225 = 144 y - 720$

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

Subtract $144 y$ from both sides of the equation.

$9 y^{2} - 90 y + 225 - 144 y = - 720$

Subtract $144 y$ from $- 90 y$ .

$9 y^{2} - 234 y + 225 = - 720$

Add $720$ to both sides of the equation.

$9 y^{2} - 234 y + 225 + 720 = 0$

Add $225$ and $720$ .

$9 y^{2} - 234 y + 945 = 0$

Factor the left side of the equation.

Factor $9$ out of $9 y^{2} - 234 y + 945$ .

Factor $9$ out of $9 y^{2}$ .

$9 (y^{2}) - 234 y + 945 = 0$

Factor $9$ out of $- 234 y$ .

$9 (y^{2}) + 9 (- 26 y) + 945 = 0$

Factor $9$ out of $945$ .

$9 y^{2} + 9 (- 26 y) + 9 \cdot 105 = 0$

Factor $9$ out of $9 y^{2} + 9 (- 26 y)$ .

$9 (y^{2} - 26 y) + 9 \cdot 105 = 0$

Factor $9$ out of $9 (y^{2} - 26 y) + 9 \cdot 105$ .

$9 (y^{2} - 26 y + 105) = 0$

Factor.

Factor $y^{2} - 26 y + 105$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $105$ and whose sum is $- 26$ .

$- 21, - 5$

Write the factored form using these integers.

$9 ((y - 21) (y - 5)) = 0$

Remove unnecessary parentheses.

$9 (y - 21) (y - 5) = 0$

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

$y - 21 = 0$

$y - 5 = 0$

Set $y - 21$ equal to $0$ and solve for $y$ .

Set $y - 21$ equal to $0$ .

$y - 21 = 0$

Add $21$ to both sides of the equation.

$y = 21$

Set $y - 5$ equal to $0$ and solve for $y$ .

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Add $5$ to both sides of the equation.

$y = 5$

The final solution is all the values that make $9 (y - 21) (y - 5) = 0$ true.

$y = 21, 5$

Solve for y square root of 4y+16- square root of y-5=6

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