Precalculus Examples

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

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $4 y^{2} - 16 y = x + 4$ .

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

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Subtract $x$ from both sides of the equation.

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

Step 1.2.2

Subtract $4$ from both sides of the equation.

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

Step 1.3

Use the quadratic formula to find the solutions.

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

Step 1.4

Substitute the values $a = 4$ , $b = - 16$ , and $c = - x - 4$ into the quadratic formula and solve for $y$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 4 \cdot (- x - 4)}}{2 \cdot 4}$

Step 1.5.1.2

Multiply $- 4$ by $4$ .

$y = \frac{16 \pm \sqrt{256 - 16 \cdot (- x - 4)}}{2 \cdot 4}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{16 \pm \sqrt{256 - 16 (- x) - 16 \cdot - 4}}{2 \cdot 4}$

Step 1.5.1.4

Multiply $- 1$ by $- 16$ .

$y = \frac{16 \pm \sqrt{256 + 16 x - 16 \cdot - 4}}{2 \cdot 4}$

Step 1.5.1.5

Multiply $- 16$ by $- 4$ .

$y = \frac{16 \pm \sqrt{256 + 16 x + 64}}{2 \cdot 4}$

Step 1.5.1.6

Add $256$ and $64$ .

$y = \frac{16 \pm \sqrt{16 x + 320}}{2 \cdot 4}$

Step 1.5.1.7

Factor $16$ out of $16 x + 320$ .

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Step 1.5.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{16 \pm \sqrt{16 (x) + 320}}{2 \cdot 4}$

Step 1.5.1.7.2

Factor $16$ out of $320$ .

$y = \frac{16 \pm \sqrt{16 x + 16 \cdot 20}}{2 \cdot 4}$

Step 1.5.1.7.3

Factor $16$ out of $16 x + 16 \cdot 20$ .

$y = \frac{16 \pm \sqrt{16 (x + 20)}}{2 \cdot 4}$

Step 1.5.1.8

Rewrite $16 (x + 20)$ as ${(2^{2})}^{2} (x + 20)$ .

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Step 1.5.1.8.1

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

$y = \frac{16 \pm \sqrt{4^{2} (x + 20)}}{2 \cdot 4}$

Step 1.5.1.8.2

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

$y = \frac{16 \pm \sqrt{{(2^{2})}^{2} (x + 20)}}{2 \cdot 4}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{16 \pm 2^{2} \sqrt{x + 20}}{2 \cdot 4}$

Step 1.5.1.10

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

$y = \frac{16 \pm 4 \sqrt{x + 20}}{2 \cdot 4}$

Step 1.5.2

Multiply $2$ by $4$ .

$y = \frac{16 \pm 4 \sqrt{x + 20}}{8}$

Step 1.5.3

Simplify $\frac{16 \pm 4 \sqrt{x + 20}}{8}$ .

$y = \frac{4 \pm \sqrt{x + 20}}{2}$

Step 1.6

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

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 4 \cdot (- x - 4)}}{2 \cdot 4}$

Step 1.6.1.2

Multiply $- 4$ by $4$ .

$y = \frac{16 \pm \sqrt{256 - 16 \cdot (- x - 4)}}{2 \cdot 4}$

Step 1.6.1.3

Apply the distributive property.

$y = \frac{16 \pm \sqrt{256 - 16 (- x) - 16 \cdot - 4}}{2 \cdot 4}$

Step 1.6.1.4

Multiply $- 1$ by $- 16$ .

$y = \frac{16 \pm \sqrt{256 + 16 x - 16 \cdot - 4}}{2 \cdot 4}$

Step 1.6.1.5

Multiply $- 16$ by $- 4$ .

$y = \frac{16 \pm \sqrt{256 + 16 x + 64}}{2 \cdot 4}$

Step 1.6.1.6

Add $256$ and $64$ .

$y = \frac{16 \pm \sqrt{16 x + 320}}{2 \cdot 4}$

Step 1.6.1.7

Factor $16$ out of $16 x + 320$ .

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Step 1.6.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{16 \pm \sqrt{16 (x) + 320}}{2 \cdot 4}$

Step 1.6.1.7.2

Factor $16$ out of $320$ .

$y = \frac{16 \pm \sqrt{16 x + 16 \cdot 20}}{2 \cdot 4}$

Step 1.6.1.7.3

Factor $16$ out of $16 x + 16 \cdot 20$ .

$y = \frac{16 \pm \sqrt{16 (x + 20)}}{2 \cdot 4}$

Step 1.6.1.8

Rewrite $16 (x + 20)$ as ${(2^{2})}^{2} (x + 20)$ .

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Step 1.6.1.8.1

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

$y = \frac{16 \pm \sqrt{4^{2} (x + 20)}}{2 \cdot 4}$

Step 1.6.1.8.2

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

$y = \frac{16 \pm \sqrt{{(2^{2})}^{2} (x + 20)}}{2 \cdot 4}$

Step 1.6.1.9

Pull terms out from under the radical.

$y = \frac{16 \pm 2^{2} \sqrt{x + 20}}{2 \cdot 4}$

Step 1.6.1.10

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

$y = \frac{16 \pm 4 \sqrt{x + 20}}{2 \cdot 4}$

Step 1.6.2

Multiply $2$ by $4$ .

$y = \frac{16 \pm 4 \sqrt{x + 20}}{8}$

Step 1.6.3

Simplify $\frac{16 \pm 4 \sqrt{x + 20}}{8}$ .

$y = \frac{4 \pm \sqrt{x + 20}}{2}$

Step 1.6.4

Change the $\pm$ to $+$ .

$y = \frac{4 + \sqrt{x + 20}}{2}$

Step 1.7

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

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Step 1.7.1

Simplify the numerator.

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Step 1.7.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 4 \cdot (- x - 4)}}{2 \cdot 4}$

Step 1.7.1.2

Multiply $- 4$ by $4$ .

$y = \frac{16 \pm \sqrt{256 - 16 \cdot (- x - 4)}}{2 \cdot 4}$

Step 1.7.1.3

Apply the distributive property.

$y = \frac{16 \pm \sqrt{256 - 16 (- x) - 16 \cdot - 4}}{2 \cdot 4}$

Step 1.7.1.4

Multiply $- 1$ by $- 16$ .

$y = \frac{16 \pm \sqrt{256 + 16 x - 16 \cdot - 4}}{2 \cdot 4}$

Step 1.7.1.5

Multiply $- 16$ by $- 4$ .

$y = \frac{16 \pm \sqrt{256 + 16 x + 64}}{2 \cdot 4}$

Step 1.7.1.6

Add $256$ and $64$ .

$y = \frac{16 \pm \sqrt{16 x + 320}}{2 \cdot 4}$

Step 1.7.1.7

Factor $16$ out of $16 x + 320$ .

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Step 1.7.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{16 \pm \sqrt{16 (x) + 320}}{2 \cdot 4}$

Step 1.7.1.7.2

Factor $16$ out of $320$ .

$y = \frac{16 \pm \sqrt{16 x + 16 \cdot 20}}{2 \cdot 4}$

Step 1.7.1.7.3

Factor $16$ out of $16 x + 16 \cdot 20$ .

$y = \frac{16 \pm \sqrt{16 (x + 20)}}{2 \cdot 4}$

Step 1.7.1.8

Rewrite $16 (x + 20)$ as ${(2^{2})}^{2} (x + 20)$ .

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Step 1.7.1.8.1

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

$y = \frac{16 \pm \sqrt{4^{2} (x + 20)}}{2 \cdot 4}$

Step 1.7.1.8.2

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

$y = \frac{16 \pm \sqrt{{(2^{2})}^{2} (x + 20)}}{2 \cdot 4}$

Step 1.7.1.9

Pull terms out from under the radical.

$y = \frac{16 \pm 2^{2} \sqrt{x + 20}}{2 \cdot 4}$

Step 1.7.1.10

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

$y = \frac{16 \pm 4 \sqrt{x + 20}}{2 \cdot 4}$

Step 1.7.2

Multiply $2$ by $4$ .

$y = \frac{16 \pm 4 \sqrt{x + 20}}{8}$

Step 1.7.3

Simplify $\frac{16 \pm 4 \sqrt{x + 20}}{8}$ .

$y = \frac{4 \pm \sqrt{x + 20}}{2}$

Step 1.7.4

Change the $\pm$ to $-$ .

$y = \frac{4 - \sqrt{x + 20}}{2}$

Step 1.8

The final answer is the combination of both solutions.

$y = \frac{4 + \sqrt{x + 20}}{2}$

$y = \frac{4 - \sqrt{x + 20}}{2}$

$y = \frac{4 + \sqrt{x + 20}}{2}$

$y = \frac{4 - \sqrt{x + 20}}{2}$

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 $\frac{4 + \sqrt{x + 20}}{2}$ into two fractions.

$y = \frac{4}{2} + \frac{\sqrt{x + 20}}{2}$

$y = \frac{4 - \sqrt{x + 20}}{2}$

Step 4

Divide $4$ by $2$ .

$y = 2 + \frac{\sqrt{x + 20}}{2}$

$y = \frac{4 - \sqrt{x + 20}}{2}$

Step 5

Split the fraction $\frac{4 - \sqrt{x + 20}}{2}$ into two fractions.

$y = 2 + \frac{\sqrt{x + 20}}{2}$

$y = \frac{4}{2} + \frac{- \sqrt{x + 20}}{2}$

Step 6

Simplify each term.

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Step 6.1

Divide $4$ by $2$ .

$y = 2 + \frac{\sqrt{x + 20}}{2}$

$y = 2 + \frac{- \sqrt{x + 20}}{2}$

Step 6.2

Move the negative in front of the fraction.

$y = 2 + \frac{\sqrt{x + 20}}{2}$

$y = 2 - \frac{\sqrt{x + 20}}{2}$

$y = 2 + \frac{\sqrt{x + 20}}{2}$

$y = 2 - \frac{\sqrt{x + 20}}{2}$

Step 7

Reorder terms.

$y = 2 + \frac{1}{2} \cdot \sqrt{x + 20}$

$y = 2 - (\frac{1}{2} \cdot \sqrt{x + 20})$

Step 8

Remove parentheses.

$y = 2 + \frac{1}{2} \cdot \sqrt{x + 20}$

$y = 2 - \frac{1}{2} \cdot \sqrt{x + 20}$

Step 9