Trigonometry Examples

$16 x^{2} + 4 y^{2} - 8 y - 12 = 0$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Subtract $4 y^{2}$ from both sides of the equation.

$16 x^{2} - 8 y - 12 = - 4 y^{2}$

Step 1.2

Add $8 y$ to both sides of the equation.

$16 x^{2} - 12 = - 4 y^{2} + 8 y$

Step 1.3

Add $12$ to both sides of the equation.

$16 x^{2} = - 4 y^{2} + 8 y + 12$

Step 2

Divide each term in $16 x^{2} = - 4 y^{2} + 8 y + 12$ by $16$ and simplify.

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Step 2.1

Divide each term in $16 x^{2} = - 4 y^{2} + 8 y + 12$ by $16$ .

$\frac{16 x^{2}}{16} = \frac{- 4 y^{2}}{16} + \frac{8 y}{16} + \frac{12}{16}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $16$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{16 x^{2}}{16} = \frac{- 4 y^{2}}{16} + \frac{8 y}{16} + \frac{12}{16}$

Step 2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4 y^{2}}{16} + \frac{8 y}{16} + \frac{12}{16}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Cancel the common factor of $- 4$ and $16$ .

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Step 2.3.1.1.1

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

$x^{2} = \frac{4 (- y^{2})}{16} + \frac{8 y}{16} + \frac{12}{16}$

Step 2.3.1.1.2

Cancel the common factors.

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Step 2.3.1.1.2.1

Factor $4$ out of $16$ .

$x^{2} = \frac{4 (- y^{2})}{4 (4)} + \frac{8 y}{16} + \frac{12}{16}$

Step 2.3.1.1.2.2

Cancel the common factor.

$x^{2} = \frac{4 (- y^{2})}{4 \cdot 4} + \frac{8 y}{16} + \frac{12}{16}$

Step 2.3.1.1.2.3

Rewrite the expression.

$x^{2} = \frac{- y^{2}}{4} + \frac{8 y}{16} + \frac{12}{16}$

Step 2.3.1.2

Move the negative in front of the fraction.

$x^{2} = - \frac{y^{2}}{4} + \frac{8 y}{16} + \frac{12}{16}$

Step 2.3.1.3

Cancel the common factor of $8$ and $16$ .

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Step 2.3.1.3.1

Factor $8$ out of $8 y$ .

$x^{2} = - \frac{y^{2}}{4} + \frac{8 (y)}{16} + \frac{12}{16}$

Step 2.3.1.3.2

Cancel the common factors.

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Step 2.3.1.3.2.1

Factor $8$ out of $16$ .

$x^{2} = - \frac{y^{2}}{4} + \frac{8 y}{8 \cdot 2} + \frac{12}{16}$

Step 2.3.1.3.2.2

Cancel the common factor.

$x^{2} = - \frac{y^{2}}{4} + \frac{8 y}{8 \cdot 2} + \frac{12}{16}$

Step 2.3.1.3.2.3

Rewrite the expression.

$x^{2} = - \frac{y^{2}}{4} + \frac{y}{2} + \frac{12}{16}$

Step 2.3.1.4

Cancel the common factor of $12$ and $16$ .

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Step 2.3.1.4.1

Factor $4$ out of $12$ .

$x^{2} = - \frac{y^{2}}{4} + \frac{y}{2} + \frac{4 (3)}{16}$

Step 2.3.1.4.2

Cancel the common factors.

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Step 2.3.1.4.2.1

Factor $4$ out of $16$ .

$x^{2} = - \frac{y^{2}}{4} + \frac{y}{2} + \frac{4 \cdot 3}{4 \cdot 4}$

Step 2.3.1.4.2.2

Cancel the common factor.

$x^{2} = - \frac{y^{2}}{4} + \frac{y}{2} + \frac{4 \cdot 3}{4 \cdot 4}$

Step 2.3.1.4.2.3

Rewrite the expression.

$x^{2} = - \frac{y^{2}}{4} + \frac{y}{2} + \frac{3}{4}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 4

Simplify $\pm \sqrt{- \frac{y^{2}}{4} + \frac{y}{2} + \frac{3}{4}}$ .

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Step 4.1

To write $\frac{y}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1

Multiply $\frac{y}{2}$ by $\frac{2}{2}$ .

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

Step 4.2.2

Multiply $2$ by $2$ .

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

Step 4.3

Combine into one fraction.

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Step 4.3.1

Combine the numerators over the common denominator.

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

Step 4.3.2

Combine the numerators over the common denominator.

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

Step 4.4

Move $2$ to the left of $y$ .

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

Step 4.5

Factor by grouping.

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Step 4.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 3 = - 3$ and whose sum is $b = 2$ .

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Step 4.5.1.1

Factor $2$ out of $2 y$ .

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

Step 4.5.1.2

Rewrite $2$ as $- 1$ plus $3$

$x = \pm \sqrt{\frac{- y^{2} + (- 1 + 3) y + 3}{4}}$

Step 4.5.1.3

Apply the distributive property.

$x = \pm \sqrt{\frac{- y^{2} - 1 y + 3 y + 3}{4}}$

Step 4.5.2

Factor out the greatest common factor from each group.

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Step 4.5.2.1

Group the first two terms and the last two terms.

$x = \pm \sqrt{\frac{(- y^{2} - 1 y) + 3 y + 3}{4}}$

Step 4.5.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $- y - 1$ .

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

Step 4.6

Rewrite $\frac{(- y - 1) (y - 3)}{4}$ as ${(\frac{1}{2})}^{2} ((- y - 1) (y - 3))$ .

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Step 4.6.1

Factor the perfect power $1^{2}$ out of $(- y - 1) (y - 3)$ .

$x = \pm \sqrt{\frac{1^{2} ((- y - 1) (y - 3))}{4}}$

Step 4.6.2

Factor the perfect power $2^{2}$ out of $4$ .

$x = \pm \sqrt{\frac{1^{2} ((- y - 1) (y - 3))}{2^{2} \cdot 1}}$

Step 4.6.3

Rearrange the fraction $\frac{1^{2} ((- y - 1) (y - 3))}{2^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{1}{2})}^{2} ((- y - 1) (y - 3))}$

Step 4.7

Pull terms out from under the radical.

$x = \pm \frac{1}{2} \sqrt{(- y - 1) (y - 3)}$

Step 4.8

Combine $\frac{1}{2}$ and $\sqrt{(- y - 1) (y - 3)}$ .

$x = \pm \frac{\sqrt{(- y - 1) (y - 3)}}{2}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{(- y - 1) (y - 3)}}{2}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{(- y - 1) (y - 3)}}{2}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{(- y - 1) (y - 3)}}{2}$

$x = - \frac{\sqrt{(- y - 1) (y - 3)}}{2}$

$x = \frac{\sqrt{(- y - 1) (y - 3)}}{2}$

$x = - \frac{\sqrt{(- y - 1) (y - 3)}}{2}$