Finite Math Examples

$\frac{x^{2}}{2} + y^{2} - 6 x + 8 y = - 15$

Step 1

Add $15$ to both sides of the equation.

$\frac{x^{2}}{2} + y^{2} - 6 x + 8 y + 15 = 0$

Step 2

Multiply through by the least common denominator $2$ , then simplify.

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Step 2.1

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 y^{2} + 2 (- 6 x) + 2 (8 y) + 2 \cdot 15 = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 y^{2} + 2 (- 6 x) + 2 (8 y) + 2 \cdot 15 = 0$

Step 2.2.1.2

Rewrite the expression.

$x^{2} + 2 y^{2} + 2 (- 6 x) + 2 (8 y) + 2 \cdot 15 = 0$

Step 2.2.2

Multiply $- 6$ by $2$ .

$x^{2} + 2 y^{2} - 12 x + 2 (8 y) + 2 \cdot 15 = 0$

Step 2.2.3

Multiply $8$ by $2$ .

$x^{2} + 2 y^{2} - 12 x + 16 y + 2 \cdot 15 = 0$

Step 2.2.4

Multiply $2$ by $15$ .

$x^{2} + 2 y^{2} - 12 x + 16 y + 30 = 0$

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 $a = 2$ , $b = 16$ , and $c = x^{2} - 12 x + 30$ into the quadratic formula and solve for $y$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (2 \cdot (x^{2} - 12 x + 30))}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $16$ to the power of $2$ .

$y = \frac{- 16 \pm \sqrt{256 - 4 \cdot 2 \cdot (x^{2} - 12 x + 30)}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4$ by $2$ .

$y = \frac{- 16 \pm \sqrt{256 - 8 \cdot (x^{2} - 12 x + 30)}}{2 \cdot 2}$

Step 5.1.3

Apply the distributive property.

$y = \frac{- 16 \pm \sqrt{256 - 8 x^{2} - 8 (- 12 x) - 8 \cdot 30}}{2 \cdot 2}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $- 12$ by $- 8$ .

$y = \frac{- 16 \pm \sqrt{256 - 8 x^{2} + 96 x - 8 \cdot 30}}{2 \cdot 2}$

Step 5.1.4.2

Multiply $- 8$ by $30$ .

$y = \frac{- 16 \pm \sqrt{256 - 8 x^{2} + 96 x - 240}}{2 \cdot 2}$

Step 5.1.5

Subtract $240$ from $256$ .

$y = \frac{- 16 \pm \sqrt{- 8 x^{2} + 96 x + 16}}{2 \cdot 2}$

Step 5.1.6

Factor $8$ out of $- 8 x^{2} + 96 x + 16$ .

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Step 5.1.6.1

Factor $8$ out of $- 8 x^{2}$ .

$y = \frac{- 16 \pm \sqrt{8 (- x^{2}) + 96 x + 16}}{2 \cdot 2}$

Step 5.1.6.2

Factor $8$ out of $96 x$ .

$y = \frac{- 16 \pm \sqrt{8 (- x^{2}) + 8 (12 x) + 16}}{2 \cdot 2}$

Step 5.1.6.3

Factor $8$ out of $16$ .

$y = \frac{- 16 \pm \sqrt{8 (- x^{2}) + 8 (12 x) + 8 (2)}}{2 \cdot 2}$

Step 5.1.6.4

Factor $8$ out of $8 (- x^{2}) + 8 (12 x)$ .

$y = \frac{- 16 \pm \sqrt{8 (- x^{2} + 12 x) + 8 (2)}}{2 \cdot 2}$

Step 5.1.6.5

Factor $8$ out of $8 (- x^{2} + 12 x) + 8 (2)$ .

$y = \frac{- 16 \pm \sqrt{8 (- x^{2} + 12 x + 2)}}{2 \cdot 2}$

Step 5.1.7

Rewrite $8 (- x^{2} + 12 x + 2)$ as $2^{2} \cdot (2 (- x^{2} + 12 x + 2))$ .

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Step 5.1.7.1

Factor $4$ out of $8$ .

$y = \frac{- 16 \pm \sqrt{4 (2) (- x^{2} + 12 x + 2)}}{2 \cdot 2}$

Step 5.1.7.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 16 \pm \sqrt{2^{2} \cdot (2 (- x^{2} + 12 x + 2))}}{2 \cdot 2}$

Step 5.1.7.3

Add parentheses.

$y = \frac{- 16 \pm \sqrt{2^{2} \cdot (2 (- x^{2} + 12 x + 2))}}{2 \cdot 2}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{- 16 \pm 2 \sqrt{2 (- x^{2} + 12 x + 2)}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$y = \frac{- 16 \pm 2 \sqrt{2 (- x^{2} + 12 x + 2)}}{4}$

Step 5.3

Simplify $\frac{- 16 \pm 2 \sqrt{2 (- x^{2} + 12 x + 2)}}{4}$ .

$y = \frac{- 8 \pm \sqrt{2 (- x^{2} + 12 x + 2)}}{2}$

Step 6

The final answer is the combination of both solutions.

$y = - \frac{8 - \sqrt{2 (- x^{2} + 12 x + 2)}}{2}$

$y = - \frac{\sqrt{2 (- x^{2} + 12 x + 2)} + 8}{2}$