Finite Math Examples

$4 x^{2} - y^{2} + 24 x - 4 y - 68 = 0$

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 $a = - 1$ , $b = - 4$ , and $c = 4 x^{2} + 24 x - 68$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 1 \cdot (4 x^{2} + 24 x - 68))}}{2 \cdot - 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 1 \cdot (4 x^{2} + 24 x - 68)$ .

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Step 3.1.1.1

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

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 1 \cdot (4 x^{2} + 24 x - 68)}}{2 \cdot - 1}$

Step 3.1.1.2

Factor $- 4$ out of $- 4 \cdot - 1 \cdot (4 x^{2} + 24 x - 68)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 1 \cdot (4 x^{2} + 24 x - 68))}}{2 \cdot - 1}$

Step 3.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 1 \cdot (4 x^{2} + 24 x - 68))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 1 \cdot (4 x^{2} + 24 x - 68))}}{2 \cdot - 1}$

Step 3.1.2

Factor $- 1$ out of $- 4 - 1 \cdot (4 x^{2} + 24 x - 68)$ .

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Step 3.1.2.1

Reorder $- 4$ and $- 1 \cdot (4 x^{2} + 24 x - 68)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (4 x^{2} + 24 x - 68) - 4)}}{2 \cdot - 1}$

Step 3.1.2.2

Rewrite $- 4$ as $- 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (4 x^{2} + 24 x - 68) - 1 \cdot 4)}}{2 \cdot - 1}$

Step 3.1.2.3

Factor $- 1$ out of $- 1 (4 x^{2} + 24 x - 68) - 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (4 x^{2} + 24 x - 68 + 4))}}{2 \cdot - 1}$

Step 3.1.2.4

Rewrite $- 1 (4 x^{2} + 24 x - 68 + 4)$ as $- (4 x^{2} + 24 x - 68 + 4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- (4 x^{2} + 24 x - 68 + 4))}}{2 \cdot - 1}$

Step 3.1.3

Add $- 68$ and $4$ .

$y = \frac{4 \pm \sqrt{- 4 (- (4 x^{2} + 24 x - 64))}}{2 \cdot - 1}$

Step 3.1.4

Factor $4$ out of $4 x^{2} + 24 x - 64$ .

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Step 3.1.4.1

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

$y = \frac{4 \pm \sqrt{- 4 (- (4 (x^{2}) + 24 x - 64))}}{2 \cdot - 1}$

Step 3.1.4.2

Factor $4$ out of $24 x$ .

$y = \frac{4 \pm \sqrt{- 4 (- (4 (x^{2}) + 4 (6 x) - 64))}}{2 \cdot - 1}$

Step 3.1.4.3

Factor $4$ out of $- 64$ .

$y = \frac{4 \pm \sqrt{- 4 (- (4 x^{2} + 4 (6 x) + 4 \cdot - 16))}}{2 \cdot - 1}$

Step 3.1.4.4

Factor $4$ out of $4 x^{2} + 4 (6 x)$ .

$y = \frac{4 \pm \sqrt{- 4 (- (4 (x^{2} + 6 x) + 4 \cdot - 16))}}{2 \cdot - 1}$

Step 3.1.4.5

Factor $4$ out of $4 (x^{2} + 6 x) + 4 \cdot - 16$ .

$y = \frac{4 \pm \sqrt{- 4 (- (4 (x^{2} + 6 x - 16)))}}{2 \cdot - 1}$

Step 3.1.5

Factor.

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

Step 3.1.6

Multiply $- 1$ by $4$ .

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

Step 3.1.7

Multiply $- 4$ by $- 4$ .

$y = \frac{4 \pm \sqrt{16 (x - 2) (x + 8)}}{2 \cdot - 1}$

Step 3.1.8

Rewrite $16 (x - 2) (x + 8)$ as ${(2^{2})}^{2} ((x - 2) (x + 8))$ .

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Step 3.1.8.1

Rewrite $16$ as $4^{2}$ .

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

Step 3.1.8.2

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

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

Step 3.1.8.3

Add parentheses.

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

Step 3.1.9

Pull terms out from under the radical.

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

Step 3.1.10

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

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

Step 3.2

Multiply $2$ by $- 1$ .

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

Step 3.3

Simplify $\frac{4 \pm 4 \sqrt{(x - 2) (x + 8)}}{- 2}$ .

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

Step 3.4

Move the negative one from the denominator of $\frac{2 \pm 2 \sqrt{(x - 2) (x + 8)}}{- 1}$ .

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

Step 3.5

Rewrite $- 1 \cdot (2 \pm 2 \sqrt{(x - 2) (x + 8)})$ as $- (2 \pm 2 \sqrt{(x - 2) (x + 8)})$ .

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

Step 4

The final answer is the combination of both solutions.

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

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