Solve by Factoring square root of 4y+12- square root of y-6=6

$\sqrt{4 y + 12} - \sqrt{y - 6} = 6$

Step 1

Subtract $6$ from both sides of the equation.

$\sqrt{4 y + 12} - \sqrt{y - 6} - 6 = 0$

Step 2

Simplify each term.

Factor $4$ out of $4 y + 12$ .

Factor $4$ out of $4 y$ .

$\sqrt{4 (y) + 12} - \sqrt{y - 6} - 6 = 0$

Factor $4$ out of $12$ .

$\sqrt{4 y + 4 \cdot 3} - \sqrt{y - 6} - 6 = 0$

Factor $4$ out of $4 y + 4 \cdot 3$ .

$\sqrt{4 (y + 3)} - \sqrt{y - 6} - 6 = 0$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} (y + 3)} - \sqrt{y - 6} - 6 = 0$

Pull terms out from under the radical.

$2 \sqrt{y + 3} - \sqrt{y - 6} - 6 = 0$

Step 3

Move all terms not containing $2 \sqrt{y + 3}$ to the right side of the equation.

Add $\sqrt{y - 6}$ to both sides of the equation.

$2 \sqrt{y + 3} - 6 = \sqrt{y - 6}$

Add $6$ to both sides of the equation.

$2 \sqrt{y + 3} = \sqrt{y - 6} + 6$

Step 4

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

${(2 \sqrt{y + 3})}^{2} = {(\sqrt{y - 6} + 6)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y + 3}$ as ${(y + 3)}^{\frac{1}{2}}$ .

${(2 {(y + 3)}^{\frac{1}{2}})}^{2} = {(\sqrt{y - 6} + 6)}^{2}$

Simplify the left side.

Simplify ${(2 {(y + 3)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(y + 3)}^{\frac{1}{2}}$ .

$2^{2} {({(y + 3)}^{\frac{1}{2}})}^{2} = {(\sqrt{y - 6} + 6)}^{2}$

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

$4 {({(y + 3)}^{\frac{1}{2}})}^{2} = {(\sqrt{y - 6} + 6)}^{2}$

Multiply the exponents in ${({(y + 3)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(y + 3)}^{\frac{1}{2} \cdot 2} = {(\sqrt{y - 6} + 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(y + 3)}^{\frac{1}{2} \cdot 2} = {(\sqrt{y - 6} + 6)}^{2}$

Rewrite the expression.

$4 {(y + 3)}^{1} = {(\sqrt{y - 6} + 6)}^{2}$

Simplify.

$4 (y + 3) = {(\sqrt{y - 6} + 6)}^{2}$

Apply the distributive property.

$4 y + 4 \cdot 3 = {(\sqrt{y - 6} + 6)}^{2}$

Multiply $4$ by $3$ .

$4 y + 12 = {(\sqrt{y - 6} + 6)}^{2}$

Simplify the right side.

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

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

$4 y + 12 = (\sqrt{y - 6} + 6) (\sqrt{y - 6} + 6)$

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

Apply the distributive property.

$4 y + 12 = \sqrt{y - 6} (\sqrt{y - 6} + 6) + 6 (\sqrt{y - 6} + 6)$

Apply the distributive property.

$4 y + 12 = \sqrt{y - 6} \sqrt{y - 6} + \sqrt{y - 6} \cdot 6 + 6 (\sqrt{y - 6} + 6)$

Apply the distributive property.

$4 y + 12 = \sqrt{y - 6} \sqrt{y - 6} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{y - 6} \sqrt{y - 6}$ .

Raise $\sqrt{y - 6}$ to the power of $1$ .

$4 y + 12 = {\sqrt{y - 6}}^{1} \sqrt{y - 6} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

Raise $\sqrt{y - 6}$ to the power of $1$ .

$4 y + 12 = {\sqrt{y - 6}}^{1} {\sqrt{y - 6}}^{1} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

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

$4 y + 12 = {\sqrt{y - 6}}^{1 + 1} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

Add $1$ and $1$ .

$4 y + 12 = {\sqrt{y - 6}}^{2} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

Rewrite ${\sqrt{y - 6}}^{2}$ as $y - 6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y - 6}$ as ${(y - 6)}^{\frac{1}{2}}$ .

$4 y + 12 = {({(y - 6)}^{\frac{1}{2}})}^{2} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

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

$4 y + 12 = {(y - 6)}^{\frac{1}{2} \cdot 2} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

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

$4 y + 12 = {(y - 6)}^{\frac{2}{2}} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 y + 12 = {(y - 6)}^{\frac{2}{2}} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

Rewrite the expression.

$4 y + 12 = {(y - 6)}^{1} + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

Simplify.

$4 y + 12 = y - 6 + \sqrt{y - 6} \cdot 6 + 6 \sqrt{y - 6} + 6 \cdot 6$

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

$4 y + 12 = y - 6 + 6 \cdot \sqrt{y - 6} + 6 \sqrt{y - 6} + 6 \cdot 6$

Multiply $6$ by $6$ .

$4 y + 12 = y - 6 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + 36$

Add $- 6$ and $36$ .

$4 y + 12 = y + 30 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6}$

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

$4 y + 12 = y + 30 + 12 \sqrt{y - 6}$

Step 6

Solve for $12 \sqrt{y - 6}$ .

Rewrite the equation as $y + 30 + 12 \sqrt{y - 6} = 4 y + 12$ .

$y + 30 + 12 \sqrt{y - 6} = 4 y + 12$

Move all terms not containing $12 \sqrt{y - 6}$ to the right side of the equation.

Subtract $y$ from both sides of the equation.

$30 + 12 \sqrt{y - 6} = 4 y + 12 - y$

Subtract $30$ from both sides of the equation.

$12 \sqrt{y - 6} = 4 y + 12 - y - 30$

Subtract $y$ from $4 y$ .

$12 \sqrt{y - 6} = 3 y + 12 - 30$

Subtract $30$ from $12$ .

$12 \sqrt{y - 6} = 3 y - 18$

Step 7

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

${(12 \sqrt{y - 6})}^{2} = {(3 y - 18)}^{2}$

Step 8

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y - 6}$ as ${(y - 6)}^{\frac{1}{2}}$ .

${(12 {(y - 6)}^{\frac{1}{2}})}^{2} = {(3 y - 18)}^{2}$

Simplify the left side.

Simplify ${(12 {(y - 6)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $12 {(y - 6)}^{\frac{1}{2}}$ .

$12^{2} {({(y - 6)}^{\frac{1}{2}})}^{2} = {(3 y - 18)}^{2}$

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

$144 {({(y - 6)}^{\frac{1}{2}})}^{2} = {(3 y - 18)}^{2}$

Multiply the exponents in ${({(y - 6)}^{\frac{1}{2}})}^{2}$ .

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

$144 {(y - 6)}^{\frac{1}{2} \cdot 2} = {(3 y - 18)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$144 {(y - 6)}^{\frac{1}{2} \cdot 2} = {(3 y - 18)}^{2}$

Rewrite the expression.

$144 {(y - 6)}^{1} = {(3 y - 18)}^{2}$

Simplify.

$144 (y - 6) = {(3 y - 18)}^{2}$

Apply the distributive property.

$144 y + 144 \cdot - 6 = {(3 y - 18)}^{2}$

Multiply $144$ by $- 6$ .

$144 y - 864 = {(3 y - 18)}^{2}$

Simplify the right side.

Simplify ${(3 y - 18)}^{2}$ .

Rewrite ${(3 y - 18)}^{2}$ as $(3 y - 18) (3 y - 18)$ .

$144 y - 864 = (3 y - 18) (3 y - 18)$

Expand $(3 y - 18) (3 y - 18)$ using the FOIL Method.

Apply the distributive property.

$144 y - 864 = 3 y (3 y - 18) - 18 (3 y - 18)$

Apply the distributive property.

$144 y - 864 = 3 y (3 y) + 3 y \cdot - 18 - 18 (3 y - 18)$

Apply the distributive property.

$144 y - 864 = 3 y (3 y) + 3 y \cdot - 18 - 18 (3 y) - 18 \cdot - 18$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$144 y - 864 = 3 \cdot 3 y \cdot y + 3 y \cdot - 18 - 18 (3 y) - 18 \cdot - 18$

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

Move $y$ .

$144 y - 864 = 3 \cdot 3 (y \cdot y) + 3 y \cdot - 18 - 18 (3 y) - 18 \cdot - 18$

Multiply $y$ by $y$ .

$144 y - 864 = 3 \cdot 3 y^{2} + 3 y \cdot - 18 - 18 (3 y) - 18 \cdot - 18$

Multiply $3$ by $3$ .

$144 y - 864 = 9 y^{2} + 3 y \cdot - 18 - 18 (3 y) - 18 \cdot - 18$

Multiply $- 18$ by $3$ .

$144 y - 864 = 9 y^{2} - 54 y - 18 (3 y) - 18 \cdot - 18$

Multiply $3$ by $- 18$ .

$144 y - 864 = 9 y^{2} - 54 y - 54 y - 18 \cdot - 18$

Multiply $- 18$ by $- 18$ .

$144 y - 864 = 9 y^{2} - 54 y - 54 y + 324$

Subtract $54 y$ from $- 54 y$ .

$144 y - 864 = 9 y^{2} - 108 y + 324$

Step 9

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.

$9 y^{2} - 108 y + 324 = 144 y - 864$

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

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

$9 y^{2} - 108 y + 324 - 144 y = - 864$

Subtract $144 y$ from $- 108 y$ .

$9 y^{2} - 252 y + 324 = - 864$

Add $864$ to both sides of the equation.

$9 y^{2} - 252 y + 324 + 864 = 0$

Add $324$ and $864$ .

$9 y^{2} - 252 y + 1188 = 0$

Factor the left side of the equation.

Factor $9$ out of $9 y^{2} - 252 y + 1188$ .

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

$9 (y^{2}) - 252 y + 1188 = 0$

Factor $9$ out of $- 252 y$ .

$9 (y^{2}) + 9 (- 28 y) + 1188 = 0$

Factor $9$ out of $1188$ .

$9 y^{2} + 9 (- 28 y) + 9 \cdot 132 = 0$

Factor $9$ out of $9 y^{2} + 9 (- 28 y)$ .

$9 (y^{2} - 28 y) + 9 \cdot 132 = 0$

Factor $9$ out of $9 (y^{2} - 28 y) + 9 \cdot 132$ .

$9 (y^{2} - 28 y + 132) = 0$

Factor.

Factor $y^{2} - 28 y + 132$ 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 $132$ and whose sum is $- 28$ .

$- 22, - 6$

Write the factored form using these integers.

$9 ((y - 22) (y - 6)) = 0$

Remove unnecessary parentheses.

$9 (y - 22) (y - 6) = 0$

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

$y - 22 = 0$

$y - 6 = 0$

Set $y - 22$ equal to $0$ and solve for $y$ .

Set $y - 22$ equal to $0$ .

$y - 22 = 0$

Add $22$ to both sides of the equation.

$y = 22$

Set $y - 6$ equal to $0$ and solve for $y$ .

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Add $6$ to both sides of the equation.

$y = 6$

The final solution is all the values that make $9 (y - 22) (y - 6) = 0$ true.

$y = 22, 6$

Solve by Factoring square root of 4y+12- square root of y-6=6

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