Solve for y square root of 4y+12- square root of y-6=6

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 y + 12}$ as ${(4 y + 12)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

Multiply the exponents in ${({(4 y + 12)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

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

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

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

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

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

$4 y + 12 = 36 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + {\sqrt{y - 6}}^{1} \sqrt{y - 6}$

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

$4 y + 12 = 36 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + {\sqrt{y - 6}}^{1} {\sqrt{y - 6}}^{1}$

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

$4 y + 12 = 36 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + {\sqrt{y - 6}}^{1 + 1}$

Add $1$ and $1$ .

$4 y + 12 = 36 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + {\sqrt{y - 6}}^{2}$

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

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

$4 y + 12 = 36 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + {({(y - 6)}^{\frac{1}{2}})}^{2}$

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

$4 y + 12 = 36 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + {(y - 6)}^{\frac{1}{2} \cdot 2}$

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

$4 y + 12 = 36 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + {(y - 6)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 y + 12 = 36 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + {(y - 6)}^{\frac{2}{2}}$

Rewrite the expression.

$4 y + 12 = 36 + 6 \sqrt{y - 6} + 6 \sqrt{y - 6} + {(y - 6)}^{1}$

Simplify.

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

Subtract $6$ from $36$ .

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

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

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

Step 4

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

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

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

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

Subtract $30$ from both sides of the equation.

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

Subtract $y$ from both sides of the equation.

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

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 5

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 6

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 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.

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

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 for y square root of 4y+12- square root of y-6=6

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