Solve for y square root of 5y+4- square root of y+16=2

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

Step 1

Add $\sqrt{y + 16}$ to both sides of the equation.

$\sqrt{5 y + 4} = 2 + \sqrt{y + 16}$

Step 2

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

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

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 y + 4}$ as ${(5 y + 4)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(5 y + 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 y + 4)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(2 + \sqrt{y + 16})}^{2}$ .

Rewrite ${(2 + \sqrt{y + 16})}^{2}$ as $(2 + \sqrt{y + 16}) (2 + \sqrt{y + 16})$ .

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

Expand $(2 + \sqrt{y + 16}) (2 + \sqrt{y + 16})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$5 y + 4 = 2 \cdot 2 + 2 \sqrt{y + 16} + \sqrt{y + 16} (2 + \sqrt{y + 16})$

Apply the distributive property.

$5 y + 4 = 2 \cdot 2 + 2 \sqrt{y + 16} + \sqrt{y + 16} \cdot 2 + \sqrt{y + 16} \sqrt{y + 16}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$5 y + 4 = 4 + 2 \sqrt{y + 16} + \sqrt{y + 16} \cdot 2 + \sqrt{y + 16} \sqrt{y + 16}$

Move $2$ to the left of $\sqrt{y + 16}$ .

$5 y + 4 = 4 + 2 \sqrt{y + 16} + 2 \cdot \sqrt{y + 16} + \sqrt{y + 16} \sqrt{y + 16}$

Multiply $\sqrt{y + 16} \sqrt{y + 16}$ .

Raise $\sqrt{y + 16}$ to the power of $1$ .

$5 y + 4 = 4 + 2 \sqrt{y + 16} + 2 \sqrt{y + 16} + {\sqrt{y + 16}}^{1} \sqrt{y + 16}$

Raise $\sqrt{y + 16}$ to the power of $1$ .

$5 y + 4 = 4 + 2 \sqrt{y + 16} + 2 \sqrt{y + 16} + {\sqrt{y + 16}}^{1} {\sqrt{y + 16}}^{1}$

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

$5 y + 4 = 4 + 2 \sqrt{y + 16} + 2 \sqrt{y + 16} + {\sqrt{y + 16}}^{1 + 1}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{y + 16}}^{2}$ as $y + 16$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$5 y + 4 = 4 + 2 \sqrt{y + 16} + 2 \sqrt{y + 16} + y + 16$

Add $4$ and $16$ .

$5 y + 4 = 20 + 2 \sqrt{y + 16} + 2 \sqrt{y + 16} + y$

Add $2 \sqrt{y + 16}$ and $2 \sqrt{y + 16}$ .

$5 y + 4 = 20 + 4 \sqrt{y + 16} + y$

Step 4

Solve for $4 \sqrt{y + 16}$ .

Rewrite the equation as $20 + 4 \sqrt{y + 16} + y = 5 y + 4$ .

$20 + 4 \sqrt{y + 16} + y = 5 y + 4$

Move all terms not containing $4 \sqrt{y + 16}$ to the right side of the equation.

Subtract $20$ from both sides of the equation.

$4 \sqrt{y + 16} + y = 5 y + 4 - 20$

Subtract $y$ from both sides of the equation.

$4 \sqrt{y + 16} = 5 y + 4 - 20 - y$

Subtract $y$ from $5 y$ .

$4 \sqrt{y + 16} = 4 y + 4 - 20$

Subtract $20$ from $4$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $4 {(y + 16)}^{\frac{1}{2}}$ .

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

Raise $4$ to the power of $2$ .

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$16 {(y + 16)}^{1} = {(4 y - 16)}^{2}$

Simplify.

$16 (y + 16) = {(4 y - 16)}^{2}$

Apply the distributive property.

$16 y + 16 \cdot 16 = {(4 y - 16)}^{2}$

Multiply $16$ by $16$ .

$16 y + 256 = {(4 y - 16)}^{2}$

Simplify the right side.

Simplify ${(4 y - 16)}^{2}$ .

Rewrite ${(4 y - 16)}^{2}$ as $(4 y - 16) (4 y - 16)$ .

$16 y + 256 = (4 y - 16) (4 y - 16)$

Expand $(4 y - 16) (4 y - 16)$ using the FOIL Method.

Apply the distributive property.

$16 y + 256 = 4 y (4 y - 16) - 16 (4 y - 16)$

Apply the distributive property.

$16 y + 256 = 4 y (4 y) + 4 y \cdot - 16 - 16 (4 y - 16)$

Apply the distributive property.

$16 y + 256 = 4 y (4 y) + 4 y \cdot - 16 - 16 (4 y) - 16 \cdot - 16$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$16 y + 256 = 4 \cdot 4 y \cdot y + 4 y \cdot - 16 - 16 (4 y) - 16 \cdot - 16$

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

Move $y$ .

$16 y + 256 = 4 \cdot 4 (y \cdot y) + 4 y \cdot - 16 - 16 (4 y) - 16 \cdot - 16$

Multiply $y$ by $y$ .

$16 y + 256 = 4 \cdot 4 y^{2} + 4 y \cdot - 16 - 16 (4 y) - 16 \cdot - 16$

Multiply $4$ by $4$ .

$16 y + 256 = 16 y^{2} + 4 y \cdot - 16 - 16 (4 y) - 16 \cdot - 16$

Multiply $- 16$ by $4$ .

$16 y + 256 = 16 y^{2} - 64 y - 16 (4 y) - 16 \cdot - 16$

Multiply $4$ by $- 16$ .

$16 y + 256 = 16 y^{2} - 64 y - 64 y - 16 \cdot - 16$

Multiply $- 16$ by $- 16$ .

$16 y + 256 = 16 y^{2} - 64 y - 64 y + 256$

Subtract $64 y$ from $- 64 y$ .

$16 y + 256 = 16 y^{2} - 128 y + 256$

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.

$16 y^{2} - 128 y + 256 = 16 y + 256$

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

Subtract $16 y$ from both sides of the equation.

$16 y^{2} - 128 y + 256 - 16 y = 256$

Subtract $16 y$ from $- 128 y$ .

$16 y^{2} - 144 y + 256 = 256$

Subtract $256$ from both sides of the equation.

$16 y^{2} - 144 y + 256 - 256 = 0$

Combine the opposite terms in $16 y^{2} - 144 y + 256 - 256$ .

Subtract $256$ from $256$ .

$16 y^{2} - 144 y + 0 = 0$

Add $16 y^{2} - 144 y$ and $0$ .

$16 y^{2} - 144 y = 0$

Factor $16 y$ out of $16 y^{2} - 144 y$ .

Factor $16 y$ out of $16 y^{2}$ .

$16 y (y) - 144 y = 0$

Factor $16 y$ out of $- 144 y$ .

$16 y (y) + 16 y (- 9) = 0$

Factor $16 y$ out of $16 y (y) + 16 y (- 9)$ .

$16 y (y - 9) = 0$

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

$y = 0$

$y - 9 = 0$

Set $y$ equal to $0$ .

$y = 0$

Set $y - 9$ equal to $0$ and solve for $y$ .

Set $y - 9$ equal to $0$ .

$y - 9 = 0$

Add $9$ to both sides of the equation.

$y = 9$

The final solution is all the values that make $16 y (y - 9) = 0$ true.

$y = 0, 9$

Step 8

Exclude the solutions that do not make $\sqrt{5 y + 4} - \sqrt{y + 16} = 2$ true.

$y = 9$

Solve for y square root of 5y+4- square root of y+16=2

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