Pre-Algebra Examples

$2 - 20 y^{2} = 17$

Step 1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1

Subtract $2$ from both sides of the equation.

$- 20 y^{2} = 17 - 2$

Step 1.2

Subtract $2$ from $17$ .

$- 20 y^{2} = 15$

Step 2

Divide each term in $- 20 y^{2} = 15$ by $- 20$ and simplify.

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Step 2.1

Divide each term in $- 20 y^{2} = 15$ by $- 20$ .

$\frac{- 20 y^{2}}{- 20} = \frac{15}{- 20}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 20$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 20 y^{2}}{- 20} = \frac{15}{- 20}$

Step 2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{15}{- 20}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $15$ and $- 20$ .

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Step 2.3.1.1

Factor $5$ out of $15$ .

$y^{2} = \frac{5 (3)}{- 20}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $5$ out of $- 20$ .

$y^{2} = \frac{5 \cdot 3}{5 \cdot - 4}$

Step 2.3.1.2.2

Cancel the common factor.

$y^{2} = \frac{5 \cdot 3}{5 \cdot - 4}$

Step 2.3.1.2.3

Rewrite the expression.

$y^{2} = \frac{3}{- 4}$

Step 2.3.2

Move the negative in front of the fraction.

$y^{2} = - \frac{3}{4}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- \frac{3}{4}}$

Step 4

Simplify $\pm \sqrt{- \frac{3}{4}}$ .

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Step 4.1

Rewrite $- \frac{3}{4}$ as ${(\frac{i}{2})}^{2} \cdot 3$ .

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Step 4.1.1

Rewrite $- 1$ as $i^{2}$ .

$y = \pm \sqrt{i^{2} \frac{3}{4}}$

Step 4.1.2

Factor the perfect power $1^{2}$ out of $3$ .

$y = \pm \sqrt{i^{2} \frac{1^{2} \cdot 3}{4}}$

Step 4.1.3

Factor the perfect power $2^{2}$ out of $4$ .

$y = \pm \sqrt{i^{2} \frac{1^{2} \cdot 3}{2^{2} \cdot 1}}$

Step 4.1.4

Rearrange the fraction $\frac{1^{2} \cdot 3}{2^{2} \cdot 1}$ .

$y = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \cdot 3)}$

Step 4.1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

$y = \pm \sqrt{{(\frac{i}{2})}^{2} \cdot 3}$

Step 4.2

Pull terms out from under the radical.

$y = \pm \frac{i}{2} \sqrt{3}$

Step 4.3

Combine $\frac{i}{2}$ and $\sqrt{3}$ .

$y = \pm \frac{i \sqrt{3}}{2}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{i \sqrt{3}}{2}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{i \sqrt{3}}{2}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{i \sqrt{3}}{2}, - \frac{i \sqrt{3}}{2}$