Precalculus Examples

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

Step 1

Solve for $y$ .

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Step 1.1

Subtract $4 x$ from both sides of the equation.

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

Step 1.2

Use the quadratic formula to find the solutions.

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

Step 1.3

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 4 x$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot (- 4 x))}}{2 \cdot 1}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot - 4 x$ .

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Step 1.4.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Step 1.4.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot - 4 x$ .

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

Step 1.4.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot - 4 x)$ .

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

Step 1.4.1.2

Multiply $- 4$ by $1$ .

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

Step 1.4.1.3

Factor $4$ out of $- 4 - 4 x$ .

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Step 1.4.1.3.1

Factor $4$ out of $- 4$ .

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

Step 1.4.1.3.2

Factor $4$ out of $- 4 x$ .

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

Step 1.4.1.3.3

Factor $4$ out of $4 (- 1) + 4 (- x)$ .

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

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

Step 1.4.1.4

Multiply $- 4$ by $4$ .

$y = \frac{4 \pm \sqrt{- 16 (- 1 - x)}}{2 \cdot 1}$

Step 1.4.1.5

Rewrite $- 16 (- 1 - x)$ as ${(2^{2})}^{2} \cdot (- 1 (- 1 - x))$ .

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Step 1.4.1.5.1

Factor $16$ out of $- 16$ .

$y = \frac{4 \pm \sqrt{16 (- 1) (- 1 - x)}}{2 \cdot 1}$

Step 1.4.1.5.2

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

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

Step 1.4.1.5.3

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

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

Step 1.4.1.5.4

Add parentheses.

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

Step 1.4.1.6

Pull terms out from under the radical.

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

Step 1.4.1.7

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

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

Step 1.4.2

Multiply $2$ by $1$ .

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

Step 1.4.3

Simplify $\frac{4 \pm 4 \sqrt{- 1 (- 1 - x)}}{2}$ .

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

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

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot - 4 x$ .

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Step 1.5.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Step 1.5.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot - 4 x$ .

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

Step 1.5.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot - 4 x)$ .

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

Step 1.5.1.2

Multiply $- 4$ by $1$ .

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

Step 1.5.1.3

Factor $4$ out of $- 4 - 4 x$ .

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Step 1.5.1.3.1

Factor $4$ out of $- 4$ .

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

Step 1.5.1.3.2

Factor $4$ out of $- 4 x$ .

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

Step 1.5.1.3.3

Factor $4$ out of $4 (- 1) + 4 (- x)$ .

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

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

Step 1.5.1.4

Multiply $- 4$ by $4$ .

$y = \frac{4 \pm \sqrt{- 16 (- 1 - x)}}{2 \cdot 1}$

Step 1.5.1.5

Rewrite $- 16 (- 1 - x)$ as ${(2^{2})}^{2} \cdot (- 1 (- 1 - x))$ .

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Step 1.5.1.5.1

Factor $16$ out of $- 16$ .

$y = \frac{4 \pm \sqrt{16 (- 1) (- 1 - x)}}{2 \cdot 1}$

Step 1.5.1.5.2

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

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

Step 1.5.1.5.3

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

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

Step 1.5.1.5.4

Add parentheses.

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

Step 1.5.1.6

Pull terms out from under the radical.

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

Step 1.5.1.7

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

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

Step 1.5.2

Multiply $2$ by $1$ .

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

Step 1.5.3

Simplify $\frac{4 \pm 4 \sqrt{- 1 (- 1 - x)}}{2}$ .

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

Step 1.5.4

Change the $\pm$ to $+$ .

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

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

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot - 4 x$ .

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Step 1.6.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Step 1.6.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot - 4 x$ .

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

Step 1.6.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot - 4 x)$ .

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

Step 1.6.1.2

Multiply $- 4$ by $1$ .

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

Step 1.6.1.3

Factor $4$ out of $- 4 - 4 x$ .

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Step 1.6.1.3.1

Factor $4$ out of $- 4$ .

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

Step 1.6.1.3.2

Factor $4$ out of $- 4 x$ .

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

Step 1.6.1.3.3

Factor $4$ out of $4 (- 1) + 4 (- x)$ .

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

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

Step 1.6.1.4

Multiply $- 4$ by $4$ .

$y = \frac{4 \pm \sqrt{- 16 (- 1 - x)}}{2 \cdot 1}$

Step 1.6.1.5

Rewrite $- 16 (- 1 - x)$ as ${(2^{2})}^{2} \cdot (- 1 (- 1 - x))$ .

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Step 1.6.1.5.1

Factor $16$ out of $- 16$ .

$y = \frac{4 \pm \sqrt{16 (- 1) (- 1 - x)}}{2 \cdot 1}$

Step 1.6.1.5.2

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

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

Step 1.6.1.5.3

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

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

Step 1.6.1.5.4

Add parentheses.

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

Step 1.6.1.6

Pull terms out from under the radical.

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

Step 1.6.1.7

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

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

Step 1.6.2

Multiply $2$ by $1$ .

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

Step 1.6.3

Simplify $\frac{4 \pm 4 \sqrt{- 1 (- 1 - x)}}{2}$ .

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

Step 1.6.4

Change the $\pm$ to $-$ .

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

Step 1.7

The final answer is the combination of both solutions.

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

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

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

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

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

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

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

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

Step 4