Algebra Examples

$\frac{1}{5} y^{2} + \frac{1}{5} y + \frac{2}{5} = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{5}$ and $y^{2}$ .

$\frac{y^{2}}{5} + \frac{1}{5} y + \frac{2}{5} = 0$

Step 1.1.2

Combine $\frac{1}{5}$ and $y$ .

$\frac{y^{2}}{5} + \frac{y}{5} + \frac{2}{5} = 0$

Step 2

Multiply through by the least common denominator $5$ , then simplify.

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Step 2.1

Apply the distributive property.

$5 (\frac{y^{2}}{5}) + 5 (\frac{y}{5}) + 5 (\frac{2}{5}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$5 (\frac{y^{2}}{5}) + 5 (\frac{y}{5}) + 5 (\frac{2}{5}) = 0$

Step 2.2.1.2

Rewrite the expression.

$y^{2} + 5 (\frac{y}{5}) + 5 (\frac{2}{5}) = 0$

Step 2.2.2

Cancel the common factor of $5$ .

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Step 2.2.2.1

Cancel the common factor.

$y^{2} + 5 (\frac{y}{5}) + 5 (\frac{2}{5}) = 0$

Step 2.2.2.2

Rewrite the expression.

$y^{2} + y + 5 (\frac{2}{5}) = 0$

Step 2.2.3

Cancel the common factor of $5$ .

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Step 2.2.3.1

Cancel the common factor.

$y^{2} + y + 5 (\frac{2}{5}) = 0$

Step 2.2.3.2

Rewrite the expression.

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

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = 1$ , and $c = 2$ into the quadratic formula and solve for $y$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$y = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $2$ .

$y = \frac{- 1 \pm \sqrt{1 - 8}}{2 \cdot 1}$

Step 5.1.3

Subtract $8$ from $1$ .

$y = \frac{- 1 \pm \sqrt{- 7}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{- 1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$y = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 1 \pm i \sqrt{7}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{- 1 \pm i \sqrt{7}}{2}$