Algebra Examples

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

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.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $3$ by $3$ .

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

Step 3.3.1.3.1.2

Multiply $2$ by $3$ .

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

Step 3.3.1.3.1.3

Multiply $3$ by $2$ .

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

Step 3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.3.1.5

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

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Step 3.3.1.3.1.5.1

Move $y$ .

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

Step 3.3.1.3.1.5.2

Multiply $y$ by $y$ .

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

Step 3.3.1.3.1.6

Multiply $2$ by $2$ .

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

Step 3.3.1.3.2

Add $6 y$ and $6 y$ .

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

Step 4

Solve for $y$ .

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Add $y$ to both sides of the equation.

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

Step 4.2.2

Add $12 y$ and $y$ .

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

Step 4.3

Factor by grouping.

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Step 4.3.1

Reorder terms.

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

Step 4.3.2

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 $b = 13$ .

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Step 4.3.2.1

Factor $13$ out of $13 y$ .

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

Step 4.3.2.2

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

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

Step 4.3.2.3

Apply the distributive property.

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

Step 4.3.3

Factor out the greatest common factor from each group.

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Step 4.3.3.1

Group the first two terms and the last two terms.

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

Step 4.3.3.2

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

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

Step 4.3.4

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

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

Step 4.4

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

$y + 1 = 0$

$4 y + 9 = 0$

Step 4.5

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

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Step 4.5.1

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

$y + 1 = 0$

Step 4.5.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 4.6

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

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Step 4.6.1

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

$4 y + 9 = 0$

Step 4.6.2

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

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Step 4.6.2.1

Subtract $9$ from both sides of the equation.

$4 y = - 9$

Step 4.6.2.2

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

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Step 4.6.2.2.1

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

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

Step 4.6.2.2.2

Simplify the left side.

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Step 4.6.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.6.2.2.2.1.1

Cancel the common factor.

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

Step 4.6.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 4.6.2.2.3

Simplify the right side.

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Step 4.6.2.2.3.1

Move the negative in front of the fraction.

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

Step 4.7

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.

$y = - 1$