Trigonometry Examples

$3 x^{2} + 2 y^{2} - 2 x + 5 y = 11$

Step 1

Subtract $11$ from both sides of the equation.

$3 x^{2} + 2 y^{2} - 2 x + 5 y - 11 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 3$ , $b = - 2$ , and $c = 2 y^{2} + 5 y - 11$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (3 \cdot (2 y^{2} + 5 y - 11))}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot (2 y^{2} + 5 y - 11)}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot (2 y^{2} + 5 y - 11)}}{2 \cdot 3}$

Step 4.1.3

Apply the distributive property.

$x = \frac{2 \pm \sqrt{4 - 12 (2 y^{2}) - 12 (5 y) - 12 \cdot - 11}}{2 \cdot 3}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $2$ by $- 12$ .

$x = \frac{2 \pm \sqrt{4 - 24 y^{2} - 12 (5 y) - 12 \cdot - 11}}{2 \cdot 3}$

Step 4.1.4.2

Multiply $5$ by $- 12$ .

$x = \frac{2 \pm \sqrt{4 - 24 y^{2} - 60 y - 12 \cdot - 11}}{2 \cdot 3}$

Step 4.1.4.3

Multiply $- 12$ by $- 11$ .

$x = \frac{2 \pm \sqrt{4 - 24 y^{2} - 60 y + 132}}{2 \cdot 3}$

Step 4.1.5

Add $4$ and $132$ .

$x = \frac{2 \pm \sqrt{- 24 y^{2} - 60 y + 136}}{2 \cdot 3}$

Step 4.1.6

Factor $4$ out of $- 24 y^{2} - 60 y + 136$ .

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Step 4.1.6.1

Factor $4$ out of $- 24 y^{2}$ .

$x = \frac{2 \pm \sqrt{4 (- 6 y^{2}) - 60 y + 136}}{2 \cdot 3}$

Step 4.1.6.2

Factor $4$ out of $- 60 y$ .

$x = \frac{2 \pm \sqrt{4 (- 6 y^{2}) + 4 (- 15 y) + 136}}{2 \cdot 3}$

Step 4.1.6.3

Factor $4$ out of $136$ .

$x = \frac{2 \pm \sqrt{4 (- 6 y^{2}) + 4 (- 15 y) + 4 (34)}}{2 \cdot 3}$

Step 4.1.6.4

Factor $4$ out of $4 (- 6 y^{2}) + 4 (- 15 y)$ .

$x = \frac{2 \pm \sqrt{4 (- 6 y^{2} - 15 y) + 4 (34)}}{2 \cdot 3}$

Step 4.1.6.5

Factor $4$ out of $4 (- 6 y^{2} - 15 y) + 4 (34)$ .

$x = \frac{2 \pm \sqrt{4 (- 6 y^{2} - 15 y + 34)}}{2 \cdot 3}$

Step 4.1.7

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

$x = \frac{2 \pm \sqrt{2^{2} (- 6 y^{2} - 15 y + 34)}}{2 \cdot 3}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{- 6 y^{2} - 15 y + 34}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{- 6 y^{2} - 15 y + 34}}{6}$

Step 4.3

Simplify $\frac{2 \pm 2 \sqrt{- 6 y^{2} - 15 y + 34}}{6}$ .

$x = \frac{1 \pm \sqrt{- 6 y^{2} - 15 y + 34}}{3}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{- 6 y^{2} - 15 y + 34}}{3}$

$x = \frac{1 - \sqrt{- 6 y^{2} - 15 y + 34}}{3}$