Trigonometry Examples

$36 x^{2} - 216 x - 16 y^{2} - 160 y - 652 = 0$

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 $a = 36$ , $b = - 216$ , and $c = - 16 y^{2} - 160 y - 652$ into the quadratic formula and solve for $x$ .

$\frac{216 \pm \sqrt{{(- 216)}^{2} - 4 \cdot (36 \cdot (- 16 y^{2} - 160 y - 652))}}{2 \cdot 36}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 216$ to the power of $2$ .

$x = \frac{216 \pm \sqrt{46656 - 4 \cdot 36 \cdot (- 16 y^{2} - 160 y - 652)}}{2 \cdot 36}$

Step 3.1.2

Multiply $- 4$ by $36$ .

$x = \frac{216 \pm \sqrt{46656 - 144 \cdot (- 16 y^{2} - 160 y - 652)}}{2 \cdot 36}$

Step 3.1.3

Apply the distributive property.

$x = \frac{216 \pm \sqrt{46656 - 144 (- 16 y^{2}) - 144 (- 160 y) - 144 \cdot - 652}}{2 \cdot 36}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $- 16$ by $- 144$ .

$x = \frac{216 \pm \sqrt{46656 + 2304 y^{2} - 144 (- 160 y) - 144 \cdot - 652}}{2 \cdot 36}$

Step 3.1.4.2

Multiply $- 160$ by $- 144$ .

$x = \frac{216 \pm \sqrt{46656 + 2304 y^{2} + 23040 y - 144 \cdot - 652}}{2 \cdot 36}$

Step 3.1.4.3

Multiply $- 144$ by $- 652$ .

$x = \frac{216 \pm \sqrt{46656 + 2304 y^{2} + 23040 y + 93888}}{2 \cdot 36}$

Step 3.1.5

Add $46656$ and $93888$ .

$x = \frac{216 \pm \sqrt{2304 y^{2} + 23040 y + 140544}}{2 \cdot 36}$

Step 3.1.6

Factor $2304$ out of $2304 y^{2} + 23040 y + 140544$ .

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Step 3.1.6.1

Factor $2304$ out of $2304 y^{2}$ .

$x = \frac{216 \pm \sqrt{2304 (y^{2}) + 23040 y + 140544}}{2 \cdot 36}$

Step 3.1.6.2

Factor $2304$ out of $23040 y$ .

$x = \frac{216 \pm \sqrt{2304 (y^{2}) + 2304 (10 y) + 140544}}{2 \cdot 36}$

Step 3.1.6.3

Factor $2304$ out of $140544$ .

$x = \frac{216 \pm \sqrt{2304 y^{2} + 2304 (10 y) + 2304 \cdot 61}}{2 \cdot 36}$

Step 3.1.6.4

Factor $2304$ out of $2304 y^{2} + 2304 (10 y)$ .

$x = \frac{216 \pm \sqrt{2304 (y^{2} + 10 y) + 2304 \cdot 61}}{2 \cdot 36}$

Step 3.1.6.5

Factor $2304$ out of $2304 (y^{2} + 10 y) + 2304 \cdot 61$ .

$x = \frac{216 \pm \sqrt{2304 (y^{2} + 10 y + 61)}}{2 \cdot 36}$

Step 3.1.7

Rewrite $2304$ as $48^{2}$ .

$x = \frac{216 \pm \sqrt{48^{2} (y^{2} + 10 y + 61)}}{2 \cdot 36}$

Step 3.1.8

Pull terms out from under the radical.

$x = \frac{216 \pm 48 \sqrt{y^{2} + 10 y + 61}}{2 \cdot 36}$

Step 3.2

Multiply $2$ by $36$ .

$x = \frac{216 \pm 48 \sqrt{y^{2} + 10 y + 61}}{72}$

Step 3.3

Simplify $\frac{216 \pm 48 \sqrt{y^{2} + 10 y + 61}}{72}$ .

$x = \frac{9 \pm 2 \sqrt{y^{2} + 10 y + 61}}{3}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{9 + 2 \sqrt{y^{2} + 10 y + 61}}{3}$

$x = \frac{9 - 2 \sqrt{y^{2} + 10 y + 61}}{3}$