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

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

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

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

${\sqrt{5 y + 6}}^{2} = y^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 y + 6}$ as ${(5 y + 6)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

Multiply the exponents in ${({(5 y + 6)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(5 y + 6)}^{1} = y^{2}$

Simplify.

$5 y + 6 = y^{2}$

Step 4

Solve for $y$ .

Subtract $y^{2}$ from both sides of the equation.

$5 y + 6 - y^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $5 y + 6 - y^{2}$ .

Reorder the expression.

Move $6$ .

$5 y - y^{2} + 6 = 0$

Reorder $5 y$ and $- y^{2}$ .

$- y^{2} + 5 y + 6 = 0$

Factor $- 1$ out of $- y^{2}$ .

$- (y^{2}) + 5 y + 6 = 0$

Factor $- 1$ out of $5 y$ .

$- (y^{2}) - (- 5 y) + 6 = 0$

Rewrite $6$ as $- 1 (- 6)$ .

$- (y^{2}) - (- 5 y) - 1 \cdot - 6 = 0$

Factor $- 1$ out of $- (y^{2}) - (- 5 y)$ .

$- (y^{2} - 5 y) - 1 \cdot - 6 = 0$

Factor $- 1$ out of $- (y^{2} - 5 y) - 1 (- 6)$ .

$- (y^{2} - 5 y - 6) = 0$

Factor.

Factor $y^{2} - 5 y - 6$ 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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Write the factored form using these integers.

$- ((y - 6) (y + 1)) = 0$

Remove unnecessary parentheses.

$- (y - 6) (y + 1) = 0$

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

$y - 6 = 0$

$y + 1 = 0$

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.

$y = 6$

Set $y + 1$ equal to $0$ and solve for $y$ .

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Subtract $1$ from both sides of the equation.

$y = - 1$

The final solution is all the values that make $- (y - 6) (y + 1) = 0$ true.

$y = 6, - 1$

Step 5

Exclude the solutions that do not make $y = \sqrt{5 y + 6}$ true.

$y = 6$

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

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