Finite Math Examples

$3 x^{2} - 9 y + 6 = 0$

Step 1

Solve for $y$ .

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Step 1.1

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

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Step 1.1.1

Subtract $3 x^{2}$ from both sides of the equation.

$- 9 y + 6 = - 3 x^{2}$

Step 1.1.2

Subtract $6$ from both sides of the equation.

$- 9 y = - 3 x^{2} - 6$

Step 1.2

Divide each term in $- 9 y = - 3 x^{2} - 6$ by $- 9$ and simplify.

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Step 1.2.1

Divide each term in $- 9 y = - 3 x^{2} - 6$ by $- 9$ .

$\frac{- 9 y}{- 9} = \frac{- 3 x^{2}}{- 9} + \frac{- 6}{- 9}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 9 y}{- 9} = \frac{- 3 x^{2}}{- 9} + \frac{- 6}{- 9}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 x^{2}}{- 9} + \frac{- 6}{- 9}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $- 3$ and $- 9$ .

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Step 1.2.3.1.1.1

Factor $- 3$ out of $- 3 x^{2}$ .

$y = \frac{- 3 (x^{2})}{- 9} + \frac{- 6}{- 9}$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $- 3$ out of $- 9$ .

$y = \frac{- 3 (x^{2})}{- 3 (3)} + \frac{- 6}{- 9}$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{- 3 x^{2}}{- 3 \cdot 3} + \frac{- 6}{- 9}$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{3} + \frac{- 6}{- 9}$

Step 1.2.3.1.2

Cancel the common factor of $- 6$ and $- 9$ .

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Step 1.2.3.1.2.1

Factor $- 3$ out of $- 6$ .

$y = \frac{x^{2}}{3} + \frac{- 3 (2)}{- 9}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $- 3$ out of $- 9$ .

$y = \frac{x^{2}}{3} + \frac{- 3 \cdot 2}{- 3 \cdot 3}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{x^{2}}{3} + \frac{- 3 \cdot 2}{- 3 \cdot 3}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{x^{2}}{3} + \frac{2}{3}$

Step 2

Set $0$ equal to $0$ .

$\frac{x^{2}}{3} + \frac{2}{3} = 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $\frac{2}{3}$ from both sides of the equation.

$\frac{x^{2}}{3} = - \frac{2}{3}$

Step 3.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x^{2} = - 2$

Step 3.3

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

$x = \pm \sqrt{- 2}$

Step 3.4

Simplify $\pm \sqrt{- 2}$ .

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Step 3.4.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 3.4.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 3.4.3

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

$x = \pm i \sqrt{2}$

Step 3.5

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

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Step 3.5.1

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

$x = i \sqrt{2}$

Step 3.5.2

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

$x = - i \sqrt{2}$

Step 3.5.3

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

$x = i \sqrt{2}, - i \sqrt{2}$

Step 4