Trigonometry Examples

$4 y^{2} - 10 x^{2} - 40 x - 8 y - 76 = 0$

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 $a = 4$ , $b = - 8$ , and $c = - 10 x^{2} - 40 x - 76$ into the quadratic formula and solve for $y$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (4 \cdot (- 10 x^{2} - 40 x - 76))}}{2 \cdot 4}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Raise $- 8$ to the power of $2$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 4 \cdot (- 10 x^{2} - 40 x - 76)}}{2 \cdot 4}$

Step 3.1.2

Multiply $- 4$ by $4$ .

$y = \frac{8 \pm \sqrt{64 - 16 \cdot (- 10 x^{2} - 40 x - 76)}}{2 \cdot 4}$

Step 3.1.3

Apply the distributive property.

$y = \frac{8 \pm \sqrt{64 - 16 (- 10 x^{2}) - 16 (- 40 x) - 16 \cdot - 76}}{2 \cdot 4}$

Step 3.1.4

Simplify.

Tap for more steps...

Step 3.1.4.1

Multiply $- 10$ by $- 16$ .

$y = \frac{8 \pm \sqrt{64 + 160 x^{2} - 16 (- 40 x) - 16 \cdot - 76}}{2 \cdot 4}$

Step 3.1.4.2

Multiply $- 40$ by $- 16$ .

$y = \frac{8 \pm \sqrt{64 + 160 x^{2} + 640 x - 16 \cdot - 76}}{2 \cdot 4}$

Step 3.1.4.3

Multiply $- 16$ by $- 76$ .

$y = \frac{8 \pm \sqrt{64 + 160 x^{2} + 640 x + 1216}}{2 \cdot 4}$

Step 3.1.5

Add $64$ and $1216$ .

$y = \frac{8 \pm \sqrt{160 x^{2} + 640 x + 1280}}{2 \cdot 4}$

Step 3.1.6

Factor $160$ out of $160 x^{2} + 640 x + 1280$ .

Tap for more steps...

Step 3.1.6.1

Factor $160$ out of $160 x^{2}$ .

$y = \frac{8 \pm \sqrt{160 (x^{2}) + 640 x + 1280}}{2 \cdot 4}$

Step 3.1.6.2

Factor $160$ out of $640 x$ .

$y = \frac{8 \pm \sqrt{160 (x^{2}) + 160 (4 x) + 1280}}{2 \cdot 4}$

Step 3.1.6.3

Factor $160$ out of $1280$ .

$y = \frac{8 \pm \sqrt{160 x^{2} + 160 (4 x) + 160 \cdot 8}}{2 \cdot 4}$

Step 3.1.6.4

Factor $160$ out of $160 x^{2} + 160 (4 x)$ .

$y = \frac{8 \pm \sqrt{160 (x^{2} + 4 x) + 160 \cdot 8}}{2 \cdot 4}$

Step 3.1.6.5

Factor $160$ out of $160 (x^{2} + 4 x) + 160 \cdot 8$ .

$y = \frac{8 \pm \sqrt{160 (x^{2} + 4 x + 8)}}{2 \cdot 4}$

Step 3.1.7

Rewrite $160 (x^{2} + 4 x + 8)$ as ${(2^{2})}^{2} \cdot (10 (x^{2} + 4 x + 8))$ .

Tap for more steps...

Step 3.1.7.1

Factor $16$ out of $160$ .

$y = \frac{8 \pm \sqrt{16 (10) (x^{2} + 4 x + 8)}}{2 \cdot 4}$

Step 3.1.7.2

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

$y = \frac{8 \pm \sqrt{4^{2} \cdot (10 (x^{2} + 4 x + 8))}}{2 \cdot 4}$

Step 3.1.7.3

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

$y = \frac{8 \pm \sqrt{{(2^{2})}^{2} \cdot (10 (x^{2} + 4 x + 8))}}{2 \cdot 4}$

Step 3.1.7.4

Add parentheses.

$y = \frac{8 \pm \sqrt{{(2^{2})}^{2} \cdot (10 (x^{2} + 4 x + 8))}}{2 \cdot 4}$

Step 3.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm 2^{2} \sqrt{10 (x^{2} + 4 x + 8)}}{2 \cdot 4}$

Step 3.1.9

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

$y = \frac{8 \pm 4 \sqrt{10 (x^{2} + 4 x + 8)}}{2 \cdot 4}$

Step 3.2

Multiply $2$ by $4$ .

$y = \frac{8 \pm 4 \sqrt{10 (x^{2} + 4 x + 8)}}{8}$

Step 3.3

Simplify $\frac{8 \pm 4 \sqrt{10 (x^{2} + 4 x + 8)}}{8}$ .

$y = \frac{2 \pm \sqrt{10 (x^{2} + 4 x + 8)}}{2}$

Step 4

The final answer is the combination of both solutions.

$y = \frac{2 + \sqrt{10 (x^{2} + 4 x + 8)}}{2}$

$y = \frac{2 - \sqrt{10 (x^{2} + 4 x + 8)}}{2}$