Finite Math Examples

$x^{2} - 2 x - 14 + y^{2} - 2 y = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 2$ , and $c = x^{2} - 2 x - 14$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot (x^{2} - 2 x - 14))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot (x^{2} - 2 x - 14)}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $1$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot (x^{2} - 2 x - 14)}}{2 \cdot 1}$

Step 3.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 - 4 x^{2} - 4 (- 2 x) - 4 \cdot - 14}}{2 \cdot 1}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $- 2$ by $- 4$ .

$y = \frac{2 \pm \sqrt{4 - 4 x^{2} + 8 x - 4 \cdot - 14}}{2 \cdot 1}$

Step 3.1.4.2

Multiply $- 4$ by $- 14$ .

$y = \frac{2 \pm \sqrt{4 - 4 x^{2} + 8 x + 56}}{2 \cdot 1}$

Step 3.1.5

Add $4$ and $56$ .

$y = \frac{2 \pm \sqrt{- 4 x^{2} + 8 x + 60}}{2 \cdot 1}$

Step 3.1.6

Rewrite $- 4 x^{2} + 8 x + 60$ in a factored form.

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Step 3.1.6.1

Factor $4$ out of $- 4 x^{2} + 8 x + 60$ .

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Step 3.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{2 \pm \sqrt{4 (- x^{2}) + 8 x + 60}}{2 \cdot 1}$

Step 3.1.6.1.2

Factor $4$ out of $8 x$ .

$y = \frac{2 \pm \sqrt{4 (- x^{2}) + 4 (2 x) + 60}}{2 \cdot 1}$

Step 3.1.6.1.3

Factor $4$ out of $60$ .

$y = \frac{2 \pm \sqrt{4 (- x^{2}) + 4 (2 x) + 4 (15)}}{2 \cdot 1}$

Step 3.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (2 x)$ .

$y = \frac{2 \pm \sqrt{4 (- x^{2} + 2 x) + 4 (15)}}{2 \cdot 1}$

Step 3.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 2 x) + 4 (15)$ .

$y = \frac{2 \pm \sqrt{4 (- x^{2} + 2 x + 15)}}{2 \cdot 1}$

Step 3.1.6.2

Factor by grouping.

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Step 3.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 15 = - 15$ and whose sum is $b = 2$ .

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Step 3.1.6.2.1.1

Factor $2$ out of $2 x$ .

$y = \frac{2 \pm \sqrt{4 (- x^{2} + 2 (x) + 15)}}{2 \cdot 1}$

Step 3.1.6.2.1.2

Rewrite $2$ as $- 3$ plus $5$

$y = \frac{2 \pm \sqrt{4 (- x^{2} + (- 3 + 5) x + 15)}}{2 \cdot 1}$

Step 3.1.6.2.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 (- x^{2} - 3 x + 5 x + 15)}}{2 \cdot 1}$

Step 3.1.6.2.2

Factor out the greatest common factor from each group.

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Step 3.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{2 \pm \sqrt{4 ((- x^{2} - 3 x) + 5 x + 15)}}{2 \cdot 1}$

Step 3.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 3$ .

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

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

Step 3.1.7

Rewrite $4 (- x - 3) (x - 5)$ as $2^{2} ((- x - 3) (x - 5))$ .

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Step 3.1.7.1

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

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

Step 3.1.7.2

Add parentheses.

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

Step 3.1.8

Pull terms out from under the radical.

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

Step 3.2

Multiply $2$ by $1$ .

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

Step 3.3

Simplify $\frac{2 \pm 2 \sqrt{(- x - 3) (x - 5)}}{2}$ .

$y = 1 \pm \sqrt{(- x - 3) (x - 5)}$

Step 4

The final answer is the combination of both solutions.

$y = 1 + \sqrt{(- x - 3) (x - 5)}$

$y = 1 - \sqrt{(- x - 3) (x - 5)}$