Solve by Factoring square root of 4y+29- square root of y-4=6

$\sqrt{4 y + 29} - \sqrt{y - 4} = 6$

Step 1

Subtract $6$ from both sides of the equation.

$\sqrt{4 y + 29} - \sqrt{y - 4} - 6 = 0$

Step 2

Move all terms not containing $\sqrt{4 y + 29}$ to the right side of the equation.

Add $\sqrt{y - 4}$ to both sides of the equation.

$\sqrt{4 y + 29} - 6 = \sqrt{y - 4}$

Add $6$ to both sides of the equation.

$\sqrt{4 y + 29} = \sqrt{y - 4} + 6$

Step 3

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

${\sqrt{4 y + 29}}^{2} = {(\sqrt{y - 4} + 6)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 y + 29}$ as ${(4 y + 29)}^{\frac{1}{2}}$ .

${({(4 y + 29)}^{\frac{1}{2}})}^{2} = {(\sqrt{y - 4} + 6)}^{2}$

Simplify the left side.

Simplify ${({(4 y + 29)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 y + 29)}^{\frac{1}{2}})}^{2}$ .

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

${(4 y + 29)}^{\frac{1}{2} \cdot 2} = {(\sqrt{y - 4} + 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 y + 29)}^{\frac{1}{2} \cdot 2} = {(\sqrt{y - 4} + 6)}^{2}$

Rewrite the expression.

${(4 y + 29)}^{1} = {(\sqrt{y - 4} + 6)}^{2}$

Simplify.

$4 y + 29 = {(\sqrt{y - 4} + 6)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{y - 4} + 6)}^{2}$ .

Rewrite ${(\sqrt{y - 4} + 6)}^{2}$ as $(\sqrt{y - 4} + 6) (\sqrt{y - 4} + 6)$ .

$4 y + 29 = (\sqrt{y - 4} + 6) (\sqrt{y - 4} + 6)$

Expand $(\sqrt{y - 4} + 6) (\sqrt{y - 4} + 6)$ using the FOIL Method.

Apply the distributive property.

$4 y + 29 = \sqrt{y - 4} (\sqrt{y - 4} + 6) + 6 (\sqrt{y - 4} + 6)$

Apply the distributive property.

$4 y + 29 = \sqrt{y - 4} \sqrt{y - 4} + \sqrt{y - 4} \cdot 6 + 6 (\sqrt{y - 4} + 6)$

Apply the distributive property.

$4 y + 29 = \sqrt{y - 4} \sqrt{y - 4} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{y - 4} \sqrt{y - 4}$ .

Raise $\sqrt{y - 4}$ to the power of $1$ .

$4 y + 29 = {\sqrt{y - 4}}^{1} \sqrt{y - 4} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

Raise $\sqrt{y - 4}$ to the power of $1$ .

$4 y + 29 = {\sqrt{y - 4}}^{1} {\sqrt{y - 4}}^{1} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

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

$4 y + 29 = {\sqrt{y - 4}}^{1 + 1} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

Add $1$ and $1$ .

$4 y + 29 = {\sqrt{y - 4}}^{2} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

Rewrite ${\sqrt{y - 4}}^{2}$ as $y - 4$ .

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

$4 y + 29 = {({(y - 4)}^{\frac{1}{2}})}^{2} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

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

$4 y + 29 = {(y - 4)}^{\frac{1}{2} \cdot 2} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

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

$4 y + 29 = {(y - 4)}^{\frac{2}{2}} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 y + 29 = {(y - 4)}^{\frac{2}{2}} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

Rewrite the expression.

$4 y + 29 = {(y - 4)}^{1} + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

Simplify.

$4 y + 29 = y - 4 + \sqrt{y - 4} \cdot 6 + 6 \sqrt{y - 4} + 6 \cdot 6$

Move $6$ to the left of $\sqrt{y - 4}$ .

$4 y + 29 = y - 4 + 6 \cdot \sqrt{y - 4} + 6 \sqrt{y - 4} + 6 \cdot 6$

Multiply $6$ by $6$ .

$4 y + 29 = y - 4 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + 36$

Add $- 4$ and $36$ .

$4 y + 29 = y + 32 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4}$

Add $6 \sqrt{y - 4}$ and $6 \sqrt{y - 4}$ .

$4 y + 29 = y + 32 + 12 \sqrt{y - 4}$

Step 5

Solve for $12 \sqrt{y - 4}$ .

Rewrite the equation as $y + 32 + 12 \sqrt{y - 4} = 4 y + 29$ .

$y + 32 + 12 \sqrt{y - 4} = 4 y + 29$

Move all terms not containing $12 \sqrt{y - 4}$ to the right side of the equation.

Subtract $y$ from both sides of the equation.

$32 + 12 \sqrt{y - 4} = 4 y + 29 - y$

Subtract $32$ from both sides of the equation.

$12 \sqrt{y - 4} = 4 y + 29 - y - 32$

Subtract $y$ from $4 y$ .

$12 \sqrt{y - 4} = 3 y + 29 - 32$

Subtract $32$ from $29$ .

$12 \sqrt{y - 4} = 3 y - 3$

Step 6

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

${(12 \sqrt{y - 4})}^{2} = {(3 y - 3)}^{2}$

Step 7

Simplify each side of the equation.

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

${(12 {(y - 4)}^{\frac{1}{2}})}^{2} = {(3 y - 3)}^{2}$

Simplify the left side.

Simplify ${(12 {(y - 4)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $12 {(y - 4)}^{\frac{1}{2}}$ .

$12^{2} {({(y - 4)}^{\frac{1}{2}})}^{2} = {(3 y - 3)}^{2}$

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

$144 {({(y - 4)}^{\frac{1}{2}})}^{2} = {(3 y - 3)}^{2}$

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

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

$144 {(y - 4)}^{\frac{1}{2} \cdot 2} = {(3 y - 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$144 {(y - 4)}^{\frac{1}{2} \cdot 2} = {(3 y - 3)}^{2}$

Rewrite the expression.

$144 {(y - 4)}^{1} = {(3 y - 3)}^{2}$

Simplify.

$144 (y - 4) = {(3 y - 3)}^{2}$

Apply the distributive property.

$144 y + 144 \cdot - 4 = {(3 y - 3)}^{2}$

Multiply $144$ by $- 4$ .

$144 y - 576 = {(3 y - 3)}^{2}$

Simplify the right side.

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

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

$144 y - 576 = (3 y - 3) (3 y - 3)$

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

Apply the distributive property.

$144 y - 576 = 3 y (3 y - 3) - 3 (3 y - 3)$

Apply the distributive property.

$144 y - 576 = 3 y (3 y) + 3 y \cdot - 3 - 3 (3 y - 3)$

Apply the distributive property.

$144 y - 576 = 3 y (3 y) + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$144 y - 576 = 3 \cdot 3 y \cdot y + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

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

Move $y$ .

$144 y - 576 = 3 \cdot 3 (y \cdot y) + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

Multiply $y$ by $y$ .

$144 y - 576 = 3 \cdot 3 y^{2} + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

Multiply $3$ by $3$ .

$144 y - 576 = 9 y^{2} + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

Multiply $- 3$ by $3$ .

$144 y - 576 = 9 y^{2} - 9 y - 3 (3 y) - 3 \cdot - 3$

Multiply $3$ by $- 3$ .

$144 y - 576 = 9 y^{2} - 9 y - 9 y - 3 \cdot - 3$

Multiply $- 3$ by $- 3$ .

$144 y - 576 = 9 y^{2} - 9 y - 9 y + 9$

Subtract $9 y$ from $- 9 y$ .

$144 y - 576 = 9 y^{2} - 18 y + 9$

Step 8

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.

$9 y^{2} - 18 y + 9 = 144 y - 576$

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

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

$9 y^{2} - 18 y + 9 - 144 y = - 576$

Subtract $144 y$ from $- 18 y$ .

$9 y^{2} - 162 y + 9 = - 576$

Add $576$ to both sides of the equation.

$9 y^{2} - 162 y + 9 + 576 = 0$

Add $9$ and $576$ .

$9 y^{2} - 162 y + 585 = 0$

Factor the left side of the equation.

Factor $9$ out of $9 y^{2} - 162 y + 585$ .

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

$9 (y^{2}) - 162 y + 585 = 0$

Factor $9$ out of $- 162 y$ .

$9 (y^{2}) + 9 (- 18 y) + 585 = 0$

Factor $9$ out of $585$ .

$9 y^{2} + 9 (- 18 y) + 9 \cdot 65 = 0$

Factor $9$ out of $9 y^{2} + 9 (- 18 y)$ .

$9 (y^{2} - 18 y) + 9 \cdot 65 = 0$

Factor $9$ out of $9 (y^{2} - 18 y) + 9 \cdot 65$ .

$9 (y^{2} - 18 y + 65) = 0$

Factor.

Factor $y^{2} - 18 y + 65$ 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 $65$ and whose sum is $- 18$ .

$- 13, - 5$

Write the factored form using these integers.

$9 ((y - 13) (y - 5)) = 0$

Remove unnecessary parentheses.

$9 (y - 13) (y - 5) = 0$

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

$y - 13 = 0$

$y - 5 = 0$

Set $y - 13$ equal to $0$ and solve for $y$ .

Set $y - 13$ equal to $0$ .

$y - 13 = 0$

Add $13$ to both sides of the equation.

$y = 13$

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 $9 (y - 13) (y - 5) = 0$ true.

$y = 13, 5$

Solve by Factoring square root of 4y+29- square root of y-4=6

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