Trigonometry Examples

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

Step 1

Solve for $y$ .

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

Substitute the values $a = 2$ , $b = - 3$ , and $c = 3 x$ into the quadratic formula and solve for $y$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (2 \cdot (3 x))}}{2 \cdot 2}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Raise $- 3$ to the power of $2$ .

$y = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot (3 x)}}{2 \cdot 2}$

Step 1.3.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 1.3.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{3 \pm \sqrt{9 - 8 \cdot (3 x)}}{2 \cdot 2}$

Step 1.3.1.2.2

Multiply $- 8$ by $3$ .

$y = \frac{3 \pm \sqrt{9 - 24 x}}{2 \cdot 2}$

Step 1.3.1.3

Factor $3$ out of $9 - 24 x$ .

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Step 1.3.1.3.1

Factor $3$ out of $9$ .

$y = \frac{3 \pm \sqrt{3 (3) - 24 x}}{2 \cdot 2}$

Step 1.3.1.3.2

Factor $3$ out of $- 24 x$ .

$y = \frac{3 \pm \sqrt{3 (3) + 3 (- 8 x)}}{2 \cdot 2}$

Step 1.3.1.3.3

Factor $3$ out of $3 (3) + 3 (- 8 x)$ .

$y = \frac{3 \pm \sqrt{3 (3 - 8 x)}}{2 \cdot 2}$

Step 1.3.2

Multiply $2$ by $2$ .

$y = \frac{3 \pm \sqrt{3 (3 - 8 x)}}{4}$

Step 1.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Raise $- 3$ to the power of $2$ .

$y = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot (3 x)}}{2 \cdot 2}$

Step 1.4.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 1.4.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{3 \pm \sqrt{9 - 8 \cdot (3 x)}}{2 \cdot 2}$

Step 1.4.1.2.2

Multiply $- 8$ by $3$ .

$y = \frac{3 \pm \sqrt{9 - 24 x}}{2 \cdot 2}$

Step 1.4.1.3

Factor $3$ out of $9 - 24 x$ .

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Step 1.4.1.3.1

Factor $3$ out of $9$ .

$y = \frac{3 \pm \sqrt{3 (3) - 24 x}}{2 \cdot 2}$

Step 1.4.1.3.2

Factor $3$ out of $- 24 x$ .

$y = \frac{3 \pm \sqrt{3 (3) + 3 (- 8 x)}}{2 \cdot 2}$

Step 1.4.1.3.3

Factor $3$ out of $3 (3) + 3 (- 8 x)$ .

$y = \frac{3 \pm \sqrt{3 (3 - 8 x)}}{2 \cdot 2}$

Step 1.4.2

Multiply $2$ by $2$ .

$y = \frac{3 \pm \sqrt{3 (3 - 8 x)}}{4}$

Step 1.4.3

Change the $\pm$ to $+$ .

$y = \frac{3 + \sqrt{3 (3 - 8 x)}}{4}$

Step 1.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $- 3$ to the power of $2$ .

$y = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot (3 x)}}{2 \cdot 2}$

Step 1.5.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 1.5.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{3 \pm \sqrt{9 - 8 \cdot (3 x)}}{2 \cdot 2}$

Step 1.5.1.2.2

Multiply $- 8$ by $3$ .

$y = \frac{3 \pm \sqrt{9 - 24 x}}{2 \cdot 2}$

Step 1.5.1.3

Factor $3$ out of $9 - 24 x$ .

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Step 1.5.1.3.1

Factor $3$ out of $9$ .

$y = \frac{3 \pm \sqrt{3 (3) - 24 x}}{2 \cdot 2}$

Step 1.5.1.3.2

Factor $3$ out of $- 24 x$ .

$y = \frac{3 \pm \sqrt{3 (3) + 3 (- 8 x)}}{2 \cdot 2}$

Step 1.5.1.3.3

Factor $3$ out of $3 (3) + 3 (- 8 x)$ .

$y = \frac{3 \pm \sqrt{3 (3 - 8 x)}}{2 \cdot 2}$

Step 1.5.2

Multiply $2$ by $2$ .

$y = \frac{3 \pm \sqrt{3 (3 - 8 x)}}{4}$

Step 1.5.3

Change the $\pm$ to $-$ .

$y = \frac{3 - \sqrt{3 (3 - 8 x)}}{4}$

Step 1.6

The final answer is the combination of both solutions.

$y = \frac{3 + \sqrt{3 (3 - 8 x)}}{4}$

$y = \frac{3 - \sqrt{3 (3 - 8 x)}}{4}$

$y = \frac{3 + \sqrt{3 (3 - 8 x)}}{4}$

$y = \frac{3 - \sqrt{3 (3 - 8 x)}}{4}$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

Split the fraction $\frac{3 + \sqrt{3 (3 - 8 x)}}{4}$ into two fractions.

$y = \frac{3}{4} + \frac{\sqrt{3 (3 - 8 x)}}{4}$

$y = \frac{3 - \sqrt{3 (3 - 8 x)}}{4}$

Step 4

Split the fraction $\frac{3 - \sqrt{3 (3 - 8 x)}}{4}$ into two fractions.

$y = \frac{3}{4} + \frac{\sqrt{3 (3 - 8 x)}}{4}$

$y = \frac{3}{4} + \frac{- \sqrt{3 (3 - 8 x)}}{4}$

Step 5

Move the negative in front of the fraction.

$y = \frac{3}{4} + \frac{\sqrt{3 (3 - 8 x)}}{4}$

$y = \frac{3}{4} - \frac{\sqrt{3 (3 - 8 x)}}{4}$

Step 6

Reorder terms.

$y = \frac{3}{4} + \frac{1}{4} \cdot \sqrt{3 (3 - 8 x)}$

$y = \frac{3}{4} - (\frac{1}{4} \cdot \sqrt{3 (3 - 8 x)})$

Step 7

Remove parentheses.

$y = \frac{3}{4} + \frac{1}{4} \cdot \sqrt{3 (3 - 8 x)}$

$y = \frac{3}{4} - \frac{1}{4} \cdot \sqrt{3 (3 - 8 x)}$

Step 8