Precalculus Examples

$4 y^{2} + x - 32 y + 68 = 0$

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 $a = 4$ , $b = - 32$ , and $c = x + 68$ into the quadratic formula and solve for $y$ .

$\frac{32 \pm \sqrt{{(- 32)}^{2} - 4 \cdot (4 \cdot (x + 68))}}{2 \cdot 4}$

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 $- 32$ to the power of $2$ .

$y = \frac{32 \pm \sqrt{1024 - 4 \cdot 4 \cdot (x + 68)}}{2 \cdot 4}$

Step 1.3.1.2

Multiply $- 4$ by $4$ .

$y = \frac{32 \pm \sqrt{1024 - 16 \cdot (x + 68)}}{2 \cdot 4}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{32 \pm \sqrt{1024 - 16 x - 16 \cdot 68}}{2 \cdot 4}$

Step 1.3.1.4

Multiply $- 16$ by $68$ .

$y = \frac{32 \pm \sqrt{1024 - 16 x - 1088}}{2 \cdot 4}$

Step 1.3.1.5

Subtract $1088$ from $1024$ .

$y = \frac{32 \pm \sqrt{- 16 x - 64}}{2 \cdot 4}$

Step 1.3.1.6

Factor $16$ out of $- 16 x - 64$ .

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Step 1.3.1.6.1

Factor $16$ out of $- 16 x$ .

$y = \frac{32 \pm \sqrt{16 (- x) - 64}}{2 \cdot 4}$

Step 1.3.1.6.2

Factor $16$ out of $- 64$ .

$y = \frac{32 \pm \sqrt{16 (- x) + 16 (- 4)}}{2 \cdot 4}$

Step 1.3.1.6.3

Factor $16$ out of $16 (- x) + 16 (- 4)$ .

$y = \frac{32 \pm \sqrt{16 (- x - 4)}}{2 \cdot 4}$

Step 1.3.1.7

Rewrite $16 (- x - 4)$ as ${(2^{2})}^{2} (- x - 4)$ .

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Step 1.3.1.7.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{32 \pm \sqrt{4^{2} (- x - 4)}}{2 \cdot 4}$

Step 1.3.1.7.2

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

$y = \frac{32 \pm \sqrt{{(2^{2})}^{2} (- x - 4)}}{2 \cdot 4}$

Step 1.3.1.8

Pull terms out from under the radical.

$y = \frac{32 \pm 2^{2} \sqrt{- x - 4}}{2 \cdot 4}$

Step 1.3.1.9

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

$y = \frac{32 \pm 4 \sqrt{- x - 4}}{2 \cdot 4}$

Step 1.3.2

Multiply $2$ by $4$ .

$y = \frac{32 \pm 4 \sqrt{- x - 4}}{8}$

Step 1.3.3

Simplify $\frac{32 \pm 4 \sqrt{- x - 4}}{8}$ .

$y = \frac{8 \pm \sqrt{- x - 4}}{2}$

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 $- 32$ to the power of $2$ .

$y = \frac{32 \pm \sqrt{1024 - 4 \cdot 4 \cdot (x + 68)}}{2 \cdot 4}$

Step 1.4.1.2

Multiply $- 4$ by $4$ .

$y = \frac{32 \pm \sqrt{1024 - 16 \cdot (x + 68)}}{2 \cdot 4}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{32 \pm \sqrt{1024 - 16 x - 16 \cdot 68}}{2 \cdot 4}$

Step 1.4.1.4

Multiply $- 16$ by $68$ .

$y = \frac{32 \pm \sqrt{1024 - 16 x - 1088}}{2 \cdot 4}$

Step 1.4.1.5

Subtract $1088$ from $1024$ .

$y = \frac{32 \pm \sqrt{- 16 x - 64}}{2 \cdot 4}$

Step 1.4.1.6

Factor $16$ out of $- 16 x - 64$ .

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Step 1.4.1.6.1

Factor $16$ out of $- 16 x$ .

$y = \frac{32 \pm \sqrt{16 (- x) - 64}}{2 \cdot 4}$

Step 1.4.1.6.2

Factor $16$ out of $- 64$ .

$y = \frac{32 \pm \sqrt{16 (- x) + 16 (- 4)}}{2 \cdot 4}$

Step 1.4.1.6.3

Factor $16$ out of $16 (- x) + 16 (- 4)$ .

$y = \frac{32 \pm \sqrt{16 (- x - 4)}}{2 \cdot 4}$

Step 1.4.1.7

Rewrite $16 (- x - 4)$ as ${(2^{2})}^{2} (- x - 4)$ .

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Step 1.4.1.7.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{32 \pm \sqrt{4^{2} (- x - 4)}}{2 \cdot 4}$

Step 1.4.1.7.2

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

$y = \frac{32 \pm \sqrt{{(2^{2})}^{2} (- x - 4)}}{2 \cdot 4}$

Step 1.4.1.8

Pull terms out from under the radical.

$y = \frac{32 \pm 2^{2} \sqrt{- x - 4}}{2 \cdot 4}$

Step 1.4.1.9

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

$y = \frac{32 \pm 4 \sqrt{- x - 4}}{2 \cdot 4}$

Step 1.4.2

Multiply $2$ by $4$ .

$y = \frac{32 \pm 4 \sqrt{- x - 4}}{8}$

Step 1.4.3

Simplify $\frac{32 \pm 4 \sqrt{- x - 4}}{8}$ .

$y = \frac{8 \pm \sqrt{- x - 4}}{2}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = \frac{8 + \sqrt{- x - 4}}{2}$

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 $- 32$ to the power of $2$ .

$y = \frac{32 \pm \sqrt{1024 - 4 \cdot 4 \cdot (x + 68)}}{2 \cdot 4}$

Step 1.5.1.2

Multiply $- 4$ by $4$ .

$y = \frac{32 \pm \sqrt{1024 - 16 \cdot (x + 68)}}{2 \cdot 4}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{32 \pm \sqrt{1024 - 16 x - 16 \cdot 68}}{2 \cdot 4}$

Step 1.5.1.4

Multiply $- 16$ by $68$ .

$y = \frac{32 \pm \sqrt{1024 - 16 x - 1088}}{2 \cdot 4}$

Step 1.5.1.5

Subtract $1088$ from $1024$ .

$y = \frac{32 \pm \sqrt{- 16 x - 64}}{2 \cdot 4}$

Step 1.5.1.6

Factor $16$ out of $- 16 x - 64$ .

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Step 1.5.1.6.1

Factor $16$ out of $- 16 x$ .

$y = \frac{32 \pm \sqrt{16 (- x) - 64}}{2 \cdot 4}$

Step 1.5.1.6.2

Factor $16$ out of $- 64$ .

$y = \frac{32 \pm \sqrt{16 (- x) + 16 (- 4)}}{2 \cdot 4}$

Step 1.5.1.6.3

Factor $16$ out of $16 (- x) + 16 (- 4)$ .

$y = \frac{32 \pm \sqrt{16 (- x - 4)}}{2 \cdot 4}$

Step 1.5.1.7

Rewrite $16 (- x - 4)$ as ${(2^{2})}^{2} (- x - 4)$ .

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Step 1.5.1.7.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{32 \pm \sqrt{4^{2} (- x - 4)}}{2 \cdot 4}$

Step 1.5.1.7.2

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

$y = \frac{32 \pm \sqrt{{(2^{2})}^{2} (- x - 4)}}{2 \cdot 4}$

Step 1.5.1.8

Pull terms out from under the radical.

$y = \frac{32 \pm 2^{2} \sqrt{- x - 4}}{2 \cdot 4}$

Step 1.5.1.9

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

$y = \frac{32 \pm 4 \sqrt{- x - 4}}{2 \cdot 4}$

Step 1.5.2

Multiply $2$ by $4$ .

$y = \frac{32 \pm 4 \sqrt{- x - 4}}{8}$

Step 1.5.3

Simplify $\frac{32 \pm 4 \sqrt{- x - 4}}{8}$ .

$y = \frac{8 \pm \sqrt{- x - 4}}{2}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = \frac{8 - \sqrt{- x - 4}}{2}$

Step 1.6

The final answer is the combination of both solutions.

$y = \frac{8 + \sqrt{- x - 4}}{2}$

$y = \frac{8 - \sqrt{- x - 4}}{2}$

$y = \frac{8 + \sqrt{- x - 4}}{2}$

$y = \frac{8 - \sqrt{- x - 4}}{2}$

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 $\frac{8 + \sqrt{- x - 4}}{2}$ into two fractions.

$y = \frac{8}{2} + \frac{\sqrt{- x - 4}}{2}$

$y = \frac{8 - \sqrt{- x - 4}}{2}$

Step 4

Divide $8$ by $2$ .

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

$y = \frac{8 - \sqrt{- x - 4}}{2}$

Step 5

Split the fraction $\frac{8 - \sqrt{- x - 4}}{2}$ into two fractions.

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

$y = \frac{8}{2} + \frac{- \sqrt{- x - 4}}{2}$

Step 6

Simplify each term.

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Step 6.1

Divide $8$ by $2$ .

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

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

Step 6.2

Move the negative in front of the fraction.

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

$y = 4 - \frac{\sqrt{- x - 4}}{2}$

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

$y = 4 - \frac{\sqrt{- x - 4}}{2}$

Step 7

Reorder terms.

$y = 4 + \frac{1}{2} \cdot \sqrt{- x - 4}$

$y = 4 - (\frac{1}{2} \cdot \sqrt{- x - 4})$

Step 8

Remove parentheses.

$y = 4 + \frac{1}{2} \cdot \sqrt{- x - 4}$

$y = 4 - \frac{1}{2} \cdot \sqrt{- x - 4}$

Step 9