Precalculus Examples

$4 x^{2} + 3 y^{2} - 2 x + 9 y + 16 = 0$

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 $a = 3$ , $b = 9$ , and $c = 4 x^{2} - 2 x + 16$ into the quadratic formula and solve for $y$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (3 \cdot (4 x^{2} - 2 x + 16))}}{2 \cdot 3}$

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 $9$ to the power of $2$ .

$y = \frac{- 9 \pm \sqrt{81 - 4 \cdot 3 \cdot (4 x^{2} - 2 x + 16)}}{2 \cdot 3}$

Step 1.3.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 9 \pm \sqrt{81 - 12 \cdot (4 x^{2} - 2 x + 16)}}{2 \cdot 3}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{- 9 \pm \sqrt{81 - 12 (4 x^{2}) - 12 (- 2 x) - 12 \cdot 16}}{2 \cdot 3}$

Step 1.3.1.4

Simplify.

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Step 1.3.1.4.1

Multiply $4$ by $- 12$ .

$y = \frac{- 9 \pm \sqrt{81 - 48 x^{2} - 12 (- 2 x) - 12 \cdot 16}}{2 \cdot 3}$

Step 1.3.1.4.2

Multiply $- 2$ by $- 12$ .

$y = \frac{- 9 \pm \sqrt{81 - 48 x^{2} + 24 x - 12 \cdot 16}}{2 \cdot 3}$

Step 1.3.1.4.3

Multiply $- 12$ by $16$ .

$y = \frac{- 9 \pm \sqrt{81 - 48 x^{2} + 24 x - 192}}{2 \cdot 3}$

Step 1.3.1.5

Subtract $192$ from $81$ .

$y = \frac{- 9 \pm \sqrt{- 48 x^{2} + 24 x - 111}}{2 \cdot 3}$

Step 1.3.1.6

Factor $3$ out of $- 48 x^{2} + 24 x - 111$ .

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Step 1.3.1.6.1

Factor $3$ out of $- 48 x^{2}$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2}) + 24 x - 111}}{2 \cdot 3}$

Step 1.3.1.6.2

Factor $3$ out of $24 x$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2}) + 3 (8 x) - 111}}{2 \cdot 3}$

Step 1.3.1.6.3

Factor $3$ out of $- 111$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2}) + 3 (8 x) + 3 (- 37)}}{2 \cdot 3}$

Step 1.3.1.6.4

Factor $3$ out of $3 (- 16 x^{2}) + 3 (8 x)$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2} + 8 x) + 3 (- 37)}}{2 \cdot 3}$

Step 1.3.1.6.5

Factor $3$ out of $3 (- 16 x^{2} + 8 x) + 3 (- 37)$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{2 \cdot 3}$

Step 1.3.2

Multiply $2$ by $3$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

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 $9$ to the power of $2$ .

$y = \frac{- 9 \pm \sqrt{81 - 4 \cdot 3 \cdot (4 x^{2} - 2 x + 16)}}{2 \cdot 3}$

Step 1.4.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 9 \pm \sqrt{81 - 12 \cdot (4 x^{2} - 2 x + 16)}}{2 \cdot 3}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 9 \pm \sqrt{81 - 12 (4 x^{2}) - 12 (- 2 x) - 12 \cdot 16}}{2 \cdot 3}$

Step 1.4.1.4

Simplify.

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Step 1.4.1.4.1

Multiply $4$ by $- 12$ .

$y = \frac{- 9 \pm \sqrt{81 - 48 x^{2} - 12 (- 2 x) - 12 \cdot 16}}{2 \cdot 3}$

Step 1.4.1.4.2

Multiply $- 2$ by $- 12$ .

$y = \frac{- 9 \pm \sqrt{81 - 48 x^{2} + 24 x - 12 \cdot 16}}{2 \cdot 3}$

Step 1.4.1.4.3

Multiply $- 12$ by $16$ .

$y = \frac{- 9 \pm \sqrt{81 - 48 x^{2} + 24 x - 192}}{2 \cdot 3}$

Step 1.4.1.5

Subtract $192$ from $81$ .

$y = \frac{- 9 \pm \sqrt{- 48 x^{2} + 24 x - 111}}{2 \cdot 3}$

Step 1.4.1.6

Factor $3$ out of $- 48 x^{2} + 24 x - 111$ .

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Step 1.4.1.6.1

Factor $3$ out of $- 48 x^{2}$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2}) + 24 x - 111}}{2 \cdot 3}$

Step 1.4.1.6.2

Factor $3$ out of $24 x$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2}) + 3 (8 x) - 111}}{2 \cdot 3}$

Step 1.4.1.6.3

Factor $3$ out of $- 111$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2}) + 3 (8 x) + 3 (- 37)}}{2 \cdot 3}$

Step 1.4.1.6.4

Factor $3$ out of $3 (- 16 x^{2}) + 3 (8 x)$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2} + 8 x) + 3 (- 37)}}{2 \cdot 3}$

Step 1.4.1.6.5

Factor $3$ out of $3 (- 16 x^{2} + 8 x) + 3 (- 37)$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{2 \cdot 3}$

Step 1.4.2

Multiply $2$ by $3$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

Step 1.4.3

Change the $\pm$ to $+$ .

$y = \frac{- 9 + \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

Step 1.4.4

Rewrite $- 9$ as $- 1 (9)$ .

$y = \frac{- 1 \cdot 9 + \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

Step 1.4.5

Factor $- 1$ out of $\sqrt{3 (- 16 x^{2} + 8 x - 37)}$ .

$y = \frac{- 1 \cdot 9 - 1 (- \sqrt{3 (- 16 x^{2} + 8 x - 37)})}{6}$

Step 1.4.6

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{3 (- 16 x^{2} + 8 x - 37)})$ .

$y = \frac{- 1 (9 - \sqrt{3 (- 16 x^{2} + 8 x - 37)})}{6}$

Step 1.4.7

Move the negative in front of the fraction.

$y = - \frac{9 - \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

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 $9$ to the power of $2$ .

$y = \frac{- 9 \pm \sqrt{81 - 4 \cdot 3 \cdot (4 x^{2} - 2 x + 16)}}{2 \cdot 3}$

Step 1.5.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 9 \pm \sqrt{81 - 12 \cdot (4 x^{2} - 2 x + 16)}}{2 \cdot 3}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 9 \pm \sqrt{81 - 12 (4 x^{2}) - 12 (- 2 x) - 12 \cdot 16}}{2 \cdot 3}$

Step 1.5.1.4

Simplify.

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Step 1.5.1.4.1

Multiply $4$ by $- 12$ .

$y = \frac{- 9 \pm \sqrt{81 - 48 x^{2} - 12 (- 2 x) - 12 \cdot 16}}{2 \cdot 3}$

Step 1.5.1.4.2

Multiply $- 2$ by $- 12$ .

$y = \frac{- 9 \pm \sqrt{81 - 48 x^{2} + 24 x - 12 \cdot 16}}{2 \cdot 3}$

Step 1.5.1.4.3

Multiply $- 12$ by $16$ .

$y = \frac{- 9 \pm \sqrt{81 - 48 x^{2} + 24 x - 192}}{2 \cdot 3}$

Step 1.5.1.5

Subtract $192$ from $81$ .

$y = \frac{- 9 \pm \sqrt{- 48 x^{2} + 24 x - 111}}{2 \cdot 3}$

Step 1.5.1.6

Factor $3$ out of $- 48 x^{2} + 24 x - 111$ .

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Step 1.5.1.6.1

Factor $3$ out of $- 48 x^{2}$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2}) + 24 x - 111}}{2 \cdot 3}$

Step 1.5.1.6.2

Factor $3$ out of $24 x$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2}) + 3 (8 x) - 111}}{2 \cdot 3}$

Step 1.5.1.6.3

Factor $3$ out of $- 111$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2}) + 3 (8 x) + 3 (- 37)}}{2 \cdot 3}$

Step 1.5.1.6.4

Factor $3$ out of $3 (- 16 x^{2}) + 3 (8 x)$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2} + 8 x) + 3 (- 37)}}{2 \cdot 3}$

Step 1.5.1.6.5

Factor $3$ out of $3 (- 16 x^{2} + 8 x) + 3 (- 37)$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{2 \cdot 3}$

Step 1.5.2

Multiply $2$ by $3$ .

$y = \frac{- 9 \pm \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

Step 1.5.3

Change the $\pm$ to $-$ .

$y = \frac{- 9 - \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

Step 1.5.4

Factor $- 1$ out of $- 9 - \sqrt{3 (- 16 x^{2} + 8 x - 37)}$ .

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Step 1.5.4.1

Reorder $- 9$ and $- \sqrt{3 (- 16 x^{2} + 8 x - 37)}$ .

$y = \frac{- \sqrt{3 (- 16 x^{2} + 8 x - 37)} - 9}{6}$

Step 1.5.4.2

Factor $- 1$ out of $- \sqrt{3 (- 16 x^{2} + 8 x - 37)}$ .

$y = \frac{- (\sqrt{3 (- 16 x^{2} + 8 x - 37)}) - 9}{6}$

Step 1.5.4.3

Rewrite $- 9$ as $- 1 (9)$ .

$y = \frac{- (\sqrt{3 (- 16 x^{2} + 8 x - 37)}) - 1 \cdot 9}{6}$

Step 1.5.4.4

Factor $- 1$ out of $- (\sqrt{3 (- 16 x^{2} + 8 x - 37)}) - 1 (9)$ .

$y = \frac{- (\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9)}{6}$

Step 1.5.5

Move the negative in front of the fraction.

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

Step 1.6

The final answer is the combination of both solutions.

$y = - \frac{9 - \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

$y = - \frac{9 - \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

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 $\frac{9 - \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$ into two fractions.

$y = - (\frac{9}{6} + \frac{- \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6})$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $9$ and $6$ .

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Step 4.1.1

Factor $3$ out of $9$ .

$y = - (\frac{3 (3)}{6} + \frac{- \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6})$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $3$ out of $6$ .

$y = - (\frac{3 \cdot 3}{3 \cdot 2} + \frac{- \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6})$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

Step 4.1.2.2

Cancel the common factor.

$y = - (\frac{3 \cdot 3}{3 \cdot 2} + \frac{- \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}), - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

Step 4.1.2.3

Rewrite the expression.

$y = - (\frac{3}{2} + \frac{- \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6})$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

$y = - (\frac{3}{2} + \frac{- \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6})$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

$y = - (\frac{3}{2} + \frac{- \sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6})$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

Step 4.2

Move the negative in front of the fraction.

$y = - (\frac{3}{2} - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6})$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

$y = - (\frac{3}{2} - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6})$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

Step 5

Apply the distributive property.

$y = - \frac{3}{2} + \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$

Step 6

Split the fraction $\frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)} + 9}{6}$ into two fractions.

$y = - \frac{3}{2} + \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - (\frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6} + \frac{9}{6})$

Step 7

Cancel the common factor of $9$ and $6$ .

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Step 7.1

Factor $3$ out of $9$ .

$y = - \frac{3}{2} + \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - (\frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6} + \frac{3 (3)}{6})$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $3$ out of $6$ .

$y = - \frac{3}{2} + \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - (\frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6} + \frac{3 \cdot 3}{3 \cdot 2})$

Step 7.2.2

Cancel the common factor.

$y = - \frac{3}{2} + \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}, - (\frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6} + \frac{3 \cdot 3}{3 \cdot 2})$

Step 7.2.3

Rewrite the expression.

$y = - \frac{3}{2} + \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - (\frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6} + \frac{3}{2})$

$y = - \frac{3}{2} + \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - (\frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6} + \frac{3}{2})$

$y = - \frac{3}{2} + \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - (\frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6} + \frac{3}{2})$

Step 8

Apply the distributive property.

$y = - \frac{3}{2} + \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6}$

$y = - \frac{\sqrt{3 (- 16 x^{2} + 8 x - 37)}}{6} - \frac{3}{2}$

Step 9

Reorder terms.

$y = - \frac{3}{2} + \frac{1}{6} \cdot \sqrt{3 (- 16 x^{2} + 8 x - 37)}$

$y = - (\frac{1}{6} \cdot \sqrt{3 (- 16 x^{2} + 8 x - 37)}) - \frac{3}{2}$

Step 10

Remove parentheses.

$y = - \frac{3}{2} + \frac{1}{6} \cdot \sqrt{3 (- 16 x^{2} + 8 x - 37)}$

$y = - \frac{1}{6} \cdot \sqrt{3 (- 16 x^{2} + 8 x - 37)} - \frac{3}{2}$

Step 11