Precalculus Examples

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

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 $a = 2$ , $b = 4$ , and $c = x - 8$ into the quadratic formula and solve for $y$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (2 \cdot (x - 8))}}{2 \cdot 2}$

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 $4$ to the power of $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2 \cdot (x - 8)}}{2 \cdot 2}$

Step 1.3.1.2

Multiply $- 4$ by $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 8 \cdot (x - 8)}}{2 \cdot 2}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 8 x - 8 \cdot - 8}}{2 \cdot 2}$

Step 1.3.1.4

Multiply $- 8$ by $- 8$ .

$y = \frac{- 4 \pm \sqrt{16 - 8 x + 64}}{2 \cdot 2}$

Step 1.3.1.5

Add $16$ and $64$ .

$y = \frac{- 4 \pm \sqrt{- 8 x + 80}}{2 \cdot 2}$

Step 1.3.1.6

Factor $8$ out of $- 8 x + 80$ .

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Step 1.3.1.6.1

Factor $8$ out of $- 8 x$ .

$y = \frac{- 4 \pm \sqrt{8 (- x) + 80}}{2 \cdot 2}$

Step 1.3.1.6.2

Factor $8$ out of $80$ .

$y = \frac{- 4 \pm \sqrt{8 (- x) + 8 (10)}}{2 \cdot 2}$

Step 1.3.1.6.3

Factor $8$ out of $8 (- x) + 8 (10)$ .

$y = \frac{- 4 \pm \sqrt{8 (- x + 10)}}{2 \cdot 2}$

Step 1.3.1.7

Rewrite $8 (- x + 10)$ as $2^{2} \cdot (2 (- x + 10))$ .

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Step 1.3.1.7.1

Factor $4$ out of $8$ .

$y = \frac{- 4 \pm \sqrt{4 (2) (- x + 10)}}{2 \cdot 2}$

Step 1.3.1.7.2

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

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (- x + 10))}}{2 \cdot 2}$

Step 1.3.1.7.3

Add parentheses.

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (- x + 10))}}{2 \cdot 2}$

Step 1.3.1.8

Pull terms out from under the radical.

$y = \frac{- 4 \pm 2 \sqrt{2 (- x + 10)}}{2 \cdot 2}$

Step 1.3.2

Multiply $2$ by $2$ .

$y = \frac{- 4 \pm 2 \sqrt{2 (- x + 10)}}{4}$

Step 1.3.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 (- x + 10)}}{4}$ .

$y = \frac{- 2 \pm \sqrt{2 (- x + 10)}}{2}$

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 $4$ to the power of $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2 \cdot (x - 8)}}{2 \cdot 2}$

Step 1.4.1.2

Multiply $- 4$ by $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 8 \cdot (x - 8)}}{2 \cdot 2}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 8 x - 8 \cdot - 8}}{2 \cdot 2}$

Step 1.4.1.4

Multiply $- 8$ by $- 8$ .

$y = \frac{- 4 \pm \sqrt{16 - 8 x + 64}}{2 \cdot 2}$

Step 1.4.1.5

Add $16$ and $64$ .

$y = \frac{- 4 \pm \sqrt{- 8 x + 80}}{2 \cdot 2}$

Step 1.4.1.6

Factor $8$ out of $- 8 x + 80$ .

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Step 1.4.1.6.1

Factor $8$ out of $- 8 x$ .

$y = \frac{- 4 \pm \sqrt{8 (- x) + 80}}{2 \cdot 2}$

Step 1.4.1.6.2

Factor $8$ out of $80$ .

$y = \frac{- 4 \pm \sqrt{8 (- x) + 8 (10)}}{2 \cdot 2}$

Step 1.4.1.6.3

Factor $8$ out of $8 (- x) + 8 (10)$ .

$y = \frac{- 4 \pm \sqrt{8 (- x + 10)}}{2 \cdot 2}$

Step 1.4.1.7

Rewrite $8 (- x + 10)$ as $2^{2} \cdot (2 (- x + 10))$ .

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Step 1.4.1.7.1

Factor $4$ out of $8$ .

$y = \frac{- 4 \pm \sqrt{4 (2) (- x + 10)}}{2 \cdot 2}$

Step 1.4.1.7.2

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

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (- x + 10))}}{2 \cdot 2}$

Step 1.4.1.7.3

Add parentheses.

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (- x + 10))}}{2 \cdot 2}$

Step 1.4.1.8

Pull terms out from under the radical.

$y = \frac{- 4 \pm 2 \sqrt{2 (- x + 10)}}{2 \cdot 2}$

Step 1.4.2

Multiply $2$ by $2$ .

$y = \frac{- 4 \pm 2 \sqrt{2 (- x + 10)}}{4}$

Step 1.4.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 (- x + 10)}}{4}$ .

$y = \frac{- 2 \pm \sqrt{2 (- x + 10)}}{2}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = \frac{- 2 + \sqrt{2 (- x + 10)}}{2}$

Step 1.4.5

Rewrite $- 2$ as $- 1 (2)$ .

$y = \frac{- 1 \cdot 2 + \sqrt{2 (- x + 10)}}{2}$

Step 1.4.6

Factor $- 1$ out of $\sqrt{2 (- x + 10)}$ .

$y = \frac{- 1 \cdot 2 - 1 (- \sqrt{2 (- x + 10)})}{2}$

Step 1.4.7

Factor $- 1$ out of $- 1 (2) - 1 (- \sqrt{2 (- x + 10)})$ .

$y = \frac{- 1 (2 - \sqrt{2 (- x + 10)})}{2}$

Step 1.4.8

Move the negative in front of the fraction.

$y = - \frac{2 - \sqrt{2 (- x + 10)}}{2}$

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 $4$ to the power of $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2 \cdot (x - 8)}}{2 \cdot 2}$

Step 1.5.1.2

Multiply $- 4$ by $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 8 \cdot (x - 8)}}{2 \cdot 2}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 8 x - 8 \cdot - 8}}{2 \cdot 2}$

Step 1.5.1.4

Multiply $- 8$ by $- 8$ .

$y = \frac{- 4 \pm \sqrt{16 - 8 x + 64}}{2 \cdot 2}$

Step 1.5.1.5

Add $16$ and $64$ .

$y = \frac{- 4 \pm \sqrt{- 8 x + 80}}{2 \cdot 2}$

Step 1.5.1.6

Factor $8$ out of $- 8 x + 80$ .

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Step 1.5.1.6.1

Factor $8$ out of $- 8 x$ .

$y = \frac{- 4 \pm \sqrt{8 (- x) + 80}}{2 \cdot 2}$

Step 1.5.1.6.2

Factor $8$ out of $80$ .

$y = \frac{- 4 \pm \sqrt{8 (- x) + 8 (10)}}{2 \cdot 2}$

Step 1.5.1.6.3

Factor $8$ out of $8 (- x) + 8 (10)$ .

$y = \frac{- 4 \pm \sqrt{8 (- x + 10)}}{2 \cdot 2}$

Step 1.5.1.7

Rewrite $8 (- x + 10)$ as $2^{2} \cdot (2 (- x + 10))$ .

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Step 1.5.1.7.1

Factor $4$ out of $8$ .

$y = \frac{- 4 \pm \sqrt{4 (2) (- x + 10)}}{2 \cdot 2}$

Step 1.5.1.7.2

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

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (- x + 10))}}{2 \cdot 2}$

Step 1.5.1.7.3

Add parentheses.

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (- x + 10))}}{2 \cdot 2}$

Step 1.5.1.8

Pull terms out from under the radical.

$y = \frac{- 4 \pm 2 \sqrt{2 (- x + 10)}}{2 \cdot 2}$

Step 1.5.2

Multiply $2$ by $2$ .

$y = \frac{- 4 \pm 2 \sqrt{2 (- x + 10)}}{4}$

Step 1.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 (- x + 10)}}{4}$ .

$y = \frac{- 2 \pm \sqrt{2 (- x + 10)}}{2}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = \frac{- 2 - \sqrt{2 (- x + 10)}}{2}$

Step 1.5.5

Factor $- 1$ out of $- 2 - \sqrt{2 (- x + 10)}$ .

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Step 1.5.5.1

Rewrite $- 2$ as $- 1 (2)$ .

$y = \frac{- 1 \cdot 2 - \sqrt{2 (- x + 10)}}{2}$

Step 1.5.5.2

Factor $- 1$ out of $- \sqrt{2 (- x + 10)}$ .

$y = \frac{- 1 \cdot 2 - (\sqrt{2 (- x + 10)})}{2}$

Step 1.5.5.3

Factor $- 1$ out of $- 1 (2) - (\sqrt{2 (- x + 10)})$ .

$y = \frac{- 1 (2 + \sqrt{2 (- x + 10)})}{2}$

Step 1.5.5.4

Rewrite $- 1 (2 + \sqrt{2 (- x + 10)})$ as $- (2 + \sqrt{2 (- x + 10)})$ .

$y = \frac{- (2 + \sqrt{2 (- x + 10)})}{2}$

Step 1.5.6

Move the negative in front of the fraction.

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

Step 1.6

The final answer is the combination of both solutions.

$y = - \frac{2 - \sqrt{2 (- x + 10)}}{2}$

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

$y = - \frac{2 - \sqrt{2 (- x + 10)}}{2}$

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

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 $\frac{2 - \sqrt{2 (- x + 10)}}{2}$ into two fractions.

$y = - (\frac{2}{2} + \frac{- \sqrt{2 (- x + 10)}}{2})$

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

Step 4

Simplify each term.

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Step 4.1

Divide $2$ by $2$ .

$y = - (1 + \frac{- \sqrt{2 (- x + 10)}}{2})$

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

Step 4.2

Move the negative in front of the fraction.

$y = - (1 - \frac{\sqrt{2 (- x + 10)}}{2})$

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

$y = - (1 - \frac{\sqrt{2 (- x + 10)}}{2})$

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

Step 5

Simplify by multiplying through.

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Step 5.1

Apply the distributive property.

$y = - 1 \cdot 1 + \frac{\sqrt{2 (- x + 10)}}{2}$

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

Step 5.2

Multiply $- 1$ by $1$ .

$y = - 1 + \frac{\sqrt{2 (- x + 10)}}{2}$

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

$y = - 1 + \frac{\sqrt{2 (- x + 10)}}{2}$

$y = - \frac{2 + \sqrt{2 (- x + 10)}}{2}$

Step 6

Split the fraction $\frac{2 + \sqrt{2 (- x + 10)}}{2}$ into two fractions.

$y = - 1 + \frac{\sqrt{2 (- x + 10)}}{2}$

$y = - (\frac{2}{2} + \frac{\sqrt{2 (- x + 10)}}{2})$

Step 7

Divide $2$ by $2$ .

$y = - 1 + \frac{\sqrt{2 (- x + 10)}}{2}$

$y = - (1 + \frac{\sqrt{2 (- x + 10)}}{2})$

Step 8

Apply the distributive property.

$y = - 1 + \frac{\sqrt{2 (- x + 10)}}{2}$

$y = - 1 \cdot 1 - \frac{\sqrt{2 (- x + 10)}}{2}$

Step 9

Multiply $- 1$ by $1$ .

$y = - 1 + \frac{\sqrt{2 (- x + 10)}}{2}$

$y = - 1 - \frac{\sqrt{2 (- x + 10)}}{2}$

Step 10

Reorder terms.

$y = - 1 + \frac{1}{2} \cdot \sqrt{2 (- x + 10)}$

$y = - 1 - (\frac{1}{2} \cdot \sqrt{2 (- x + 10)})$

Step 11

Remove parentheses.

$y = - 1 + \frac{1}{2} \cdot \sqrt{2 (- x + 10)}$

$y = - 1 - \frac{1}{2} \cdot \sqrt{2 (- x + 10)}$

Step 12