Find All Complex Solutions 3+2y = square root of -y

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

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.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$- y = (3 + 2 y) (3 + 2 y)$

Expand $(3 + 2 y) (3 + 2 y)$ using the FOIL Method.

Apply the distributive property.

$- y = 3 (3 + 2 y) + 2 y (3 + 2 y)$

Apply the distributive property.

$- y = 3 \cdot 3 + 3 (2 y) + 2 y (3 + 2 y)$

Apply the distributive property.

$- y = 3 \cdot 3 + 3 (2 y) + 2 y \cdot 3 + 2 y (2 y)$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$- y = 9 + 3 (2 y) + 2 y \cdot 3 + 2 y (2 y)$

Multiply $2$ by $3$ .

$- y = 9 + 6 y + 2 y \cdot 3 + 2 y (2 y)$

Multiply $3$ by $2$ .

$- y = 9 + 6 y + 6 y + 2 y (2 y)$

Rewrite using the commutative property of multiplication.

$- y = 9 + 6 y + 6 y + 2 \cdot 2 y \cdot y$

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

Move $y$ .

$- y = 9 + 6 y + 6 y + 2 \cdot 2 (y \cdot y)$

Multiply $y$ by $y$ .

$- y = 9 + 6 y + 6 y + 2 \cdot 2 y^{2}$

Multiply $2$ by $2$ .

$- y = 9 + 6 y + 6 y + 4 y^{2}$

Add $6 y$ and $6 y$ .

$- y = 9 + 12 y + 4 y^{2}$

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.

$9 + 12 y + 4 y^{2} = - y$

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

Add $y$ to both sides of the equation.

$9 + 12 y + 4 y^{2} + y = 0$

Add $12 y$ and $y$ .

$9 + 4 y^{2} + 13 y = 0$

Factor by grouping.

Reorder terms.

$4 y^{2} + 13 y + 9 = 0$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 9 = 36$ and whose sum is $b = 13$ .

Factor $13$ out of $13 y$ .

$4 y^{2} + 13 (y) + 9 = 0$

Rewrite $13$ as $4$ plus $9$

$4 y^{2} + (4 + 9) y + 9 = 0$

Apply the distributive property.

$4 y^{2} + 4 y + 9 y + 9 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 y^{2} + 4 y) + 9 y + 9 = 0$

Factor out the greatest common factor (GCF) from each group.

$4 y (y + 1) + 9 (y + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $y + 1$ .

$(y + 1) (4 y + 9) = 0$

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

$y + 1 = 0$

$4 y + 9 = 0$

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.

$y = - 1$

Set $4 y + 9$ equal to $0$ and solve for $y$ .

Set $4 y + 9$ equal to $0$ .

$4 y + 9 = 0$

Solve $4 y + 9 = 0$ for $y$ .

Subtract $9$ from both sides of the equation.

$4 y = - 9$

Divide each term in $4 y = - 9$ by $4$ and simplify.

Divide each term in $4 y = - 9$ by $4$ .

$\frac{4 y}{4} = \frac{- 9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 9}{4}$

Divide $y$ by $1$ .

$y = \frac{- 9}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{9}{4}$

The final solution is all the values that make $(y + 1) (4 y + 9) = 0$ true.

$y = - 1, - \frac{9}{4}$

Step 5

Exclude the solutions that do not make $3 + 2 y = \sqrt{- y}$ true.

$y = - 1$

Find All Complex Solutions 3+2y = square root of -y

Related Questions

Support LCMGCF.com by disabling your adblocker.