Finite Math Examples

$x^{2} - x y + 3 y^{2} = 5$

Step 1

Subtract $5$ from both sides of the equation.

$x^{2} - x y + 3 y^{2} - 5 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - y$ , and $c = 3 y^{2} - 5$ into the quadratic formula and solve for $x$ .

$\frac{y \pm \sqrt{{(- y)}^{2} - 4 \cdot (1 \cdot (3 y^{2} - 5))}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the product rule to $- y$ .

$x = \frac{y \pm \sqrt{{(- 1)}^{2} y^{2} - 4 \cdot 1 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 4.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{y \pm \sqrt{1 y^{2} - 4 \cdot 1 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 4.1.3

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

$x = \frac{y \pm \sqrt{y^{2} - 4 \cdot 1 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 4.1.4

Multiply $- 4$ by $1$ .

$x = \frac{y \pm \sqrt{y^{2} - 4 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 4.1.5

Apply the distributive property.

$x = \frac{y \pm \sqrt{y^{2} - 4 (3 y^{2}) - 4 \cdot - 5}}{2 \cdot 1}$

Step 4.1.6

Multiply $3$ by $- 4$ .

$x = \frac{y \pm \sqrt{y^{2} - 12 y^{2} - 4 \cdot - 5}}{2 \cdot 1}$

Step 4.1.7

Multiply $- 4$ by $- 5$ .

$x = \frac{y \pm \sqrt{y^{2} - 12 y^{2} + 20}}{2 \cdot 1}$

Step 4.1.8

Subtract $12 y^{2}$ from $y^{2}$ .

$x = \frac{y \pm \sqrt{- 11 y^{2} + 20}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{- 11 y^{2} + 20}}{2}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the product rule to $- y$ .

$x = \frac{y \pm \sqrt{{(- 1)}^{2} y^{2} - 4 \cdot 1 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 5.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{y \pm \sqrt{1 y^{2} - 4 \cdot 1 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 5.1.3

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

$x = \frac{y \pm \sqrt{y^{2} - 4 \cdot 1 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 5.1.4

Multiply $- 4$ by $1$ .

$x = \frac{y \pm \sqrt{y^{2} - 4 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 5.1.5

Apply the distributive property.

$x = \frac{y \pm \sqrt{y^{2} - 4 (3 y^{2}) - 4 \cdot - 5}}{2 \cdot 1}$

Step 5.1.6

Multiply $3$ by $- 4$ .

$x = \frac{y \pm \sqrt{y^{2} - 12 y^{2} - 4 \cdot - 5}}{2 \cdot 1}$

Step 5.1.7

Multiply $- 4$ by $- 5$ .

$x = \frac{y \pm \sqrt{y^{2} - 12 y^{2} + 20}}{2 \cdot 1}$

Step 5.1.8

Subtract $12 y^{2}$ from $y^{2}$ .

$x = \frac{y \pm \sqrt{- 11 y^{2} + 20}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{- 11 y^{2} + 20}}{2}$

Step 5.3

Change the $\pm$ to $+$ .

$x = \frac{y + \sqrt{- 11 y^{2} + 20}}{2}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the product rule to $- y$ .

$x = \frac{y \pm \sqrt{{(- 1)}^{2} y^{2} - 4 \cdot 1 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 6.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{y \pm \sqrt{1 y^{2} - 4 \cdot 1 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 6.1.3

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

$x = \frac{y \pm \sqrt{y^{2} - 4 \cdot 1 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 6.1.4

Multiply $- 4$ by $1$ .

$x = \frac{y \pm \sqrt{y^{2} - 4 \cdot (3 y^{2} - 5)}}{2 \cdot 1}$

Step 6.1.5

Apply the distributive property.

$x = \frac{y \pm \sqrt{y^{2} - 4 (3 y^{2}) - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.1.6

Multiply $3$ by $- 4$ .

$x = \frac{y \pm \sqrt{y^{2} - 12 y^{2} - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.1.7

Multiply $- 4$ by $- 5$ .

$x = \frac{y \pm \sqrt{y^{2} - 12 y^{2} + 20}}{2 \cdot 1}$

Step 6.1.8

Subtract $12 y^{2}$ from $y^{2}$ .

$x = \frac{y \pm \sqrt{- 11 y^{2} + 20}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{- 11 y^{2} + 20}}{2}$

Step 6.3

Change the $\pm$ to $-$ .

$x = \frac{y - \sqrt{- 11 y^{2} + 20}}{2}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{y + \sqrt{- 11 y^{2} + 20}}{2}$

$x = \frac{y - \sqrt{- 11 y^{2} + 20}}{2}$