Solve for y 6 square root of y-5=y+3

$6 \sqrt{y - 5} = y + 3$

Step 1

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

${(6 \sqrt{y - 5})}^{2} = {(y + 3)}^{2}$

Step 2

Simplify each side of the equation.

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

${(6 {(y - 5)}^{\frac{1}{2}})}^{2} = {(y + 3)}^{2}$

Simplify the left side.

Simplify ${(6 {(y - 5)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $6 {(y - 5)}^{\frac{1}{2}}$ .

$6^{2} {({(y - 5)}^{\frac{1}{2}})}^{2} = {(y + 3)}^{2}$

Raise $6$ to the power of $2$ .

$36 {({(y - 5)}^{\frac{1}{2}})}^{2} = {(y + 3)}^{2}$

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

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

$36 {(y - 5)}^{\frac{1}{2} \cdot 2} = {(y + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(y - 5)}^{\frac{1}{2} \cdot 2} = {(y + 3)}^{2}$

Rewrite the expression.

$36 {(y - 5)}^{1} = {(y + 3)}^{2}$

Simplify.

$36 (y - 5) = {(y + 3)}^{2}$

Apply the distributive property.

$36 y + 36 \cdot - 5 = {(y + 3)}^{2}$

Multiply $36$ by $- 5$ .

$36 y - 180 = {(y + 3)}^{2}$

Simplify the right side.

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

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

$36 y - 180 = (y + 3) (y + 3)$

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

Apply the distributive property.

$36 y - 180 = y (y + 3) + 3 (y + 3)$

Apply the distributive property.

$36 y - 180 = y \cdot y + y \cdot 3 + 3 (y + 3)$

Apply the distributive property.

$36 y - 180 = y \cdot y + y \cdot 3 + 3 y + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$36 y - 180 = y^{2} + y \cdot 3 + 3 y + 3 \cdot 3$

Move $3$ to the left of $y$ .

$36 y - 180 = y^{2} + 3 \cdot y + 3 y + 3 \cdot 3$

Multiply $3$ by $3$ .

$36 y - 180 = y^{2} + 3 y + 3 y + 9$

Add $3 y$ and $3 y$ .

$36 y - 180 = y^{2} + 6 y + 9$

Step 3

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.

$y^{2} + 6 y + 9 = 36 y - 180$

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

Subtract $36 y$ from both sides of the equation.

$y^{2} + 6 y + 9 - 36 y = - 180$

Subtract $36 y$ from $6 y$ .

$y^{2} - 30 y + 9 = - 180$

Add $180$ to both sides of the equation.

$y^{2} - 30 y + 9 + 180 = 0$

Add $9$ and $180$ .

$y^{2} - 30 y + 189 = 0$

Factor $y^{2} - 30 y + 189$ 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 $189$ and whose sum is $- 30$ .

$- 21, - 9$

Write the factored form using these integers.

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

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

$y - 21 = 0$

$y - 9 = 0$

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 $y - 9$ 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 $(y - 21) (y - 9) = 0$ true.

$y = 21, 9$

Solve for y 6 square root of y-5=y+3

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