Solve for y square root of y-2+14=y

$\sqrt{y - 2} + 14 = y$

Step 1

Subtract $14$ from both sides of the equation.

$\sqrt{y - 2} = y - 14$

Step 2

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

${\sqrt{y - 2}}^{2} = {(y - 14)}^{2}$

Step 3

Simplify each side of the equation.

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

${({(y - 2)}^{\frac{1}{2}})}^{2} = {(y - 14)}^{2}$

Simplify the left side.

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

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

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

${(y - 2)}^{\frac{1}{2} \cdot 2} = {(y - 14)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y - 2)}^{\frac{1}{2} \cdot 2} = {(y - 14)}^{2}$

Rewrite the expression.

${(y - 2)}^{1} = {(y - 14)}^{2}$

Simplify.

$y - 2 = {(y - 14)}^{2}$

Simplify the right side.

Simplify ${(y - 14)}^{2}$ .

Rewrite ${(y - 14)}^{2}$ as $(y - 14) (y - 14)$ .

$y - 2 = (y - 14) (y - 14)$

Expand $(y - 14) (y - 14)$ using the FOIL Method.

Apply the distributive property.

$y - 2 = y (y - 14) - 14 (y - 14)$

Apply the distributive property.

$y - 2 = y \cdot y + y \cdot - 14 - 14 (y - 14)$

Apply the distributive property.

$y - 2 = y \cdot y + y \cdot - 14 - 14 y - 14 \cdot - 14$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$y - 2 = y^{2} + y \cdot - 14 - 14 y - 14 \cdot - 14$

Move $- 14$ to the left of $y$ .

$y - 2 = y^{2} - 14 \cdot y - 14 y - 14 \cdot - 14$

Multiply $- 14$ by $- 14$ .

$y - 2 = y^{2} - 14 y - 14 y + 196$

Subtract $14 y$ from $- 14 y$ .

$y - 2 = y^{2} - 28 y + 196$

Step 4

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.

$y^{2} - 28 y + 196 = y - 2$

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

Subtract $y$ from both sides of the equation.

$y^{2} - 28 y + 196 - y = - 2$

Subtract $y$ from $- 28 y$ .

$y^{2} - 29 y + 196 = - 2$

Add $2$ to both sides of the equation.

$y^{2} - 29 y + 196 + 2 = 0$

Add $196$ and $2$ .

$y^{2} - 29 y + 198 = 0$

Factor $y^{2} - 29 y + 198$ 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 $198$ and whose sum is $- 29$ .

$- 18, - 11$

Write the factored form using these integers.

$(y - 18) (y - 11) = 0$

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

$y - 18 = 0$

$y - 11 = 0$

Set $y - 18$ equal to $0$ and solve for $y$ .

Set $y - 18$ equal to $0$ .

$y - 18 = 0$

Add $18$ to both sides of the equation.

$y = 18$

Set $y - 11$ equal to $0$ and solve for $y$ .

Set $y - 11$ equal to $0$ .

$y - 11 = 0$

Add $11$ to both sides of the equation.

$y = 11$

The final solution is all the values that make $(y - 18) (y - 11) = 0$ true.

$y = 18, 11$

Step 5

Exclude the solutions that do not make $\sqrt{y - 2} + 14 = y$ true.

$y = 18$

Solve for y square root of y-2+14=y

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