Precalculus Examples

$y^{2} + 4 y + 4 x + 12 = 0$

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 $a = 1$ , $b = 4$ , and $c = 4 x + 12$ into the quadratic formula and solve for $y$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot (4 x + 12))}}{2 \cdot 1}$

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 $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (4 x + 12)}}{2 \cdot 1}$

Step 1.3.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot (4 x + 12)}}{2 \cdot 1}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 4 (4 x) - 4 \cdot 12}}{2 \cdot 1}$

Step 1.3.1.4

Multiply $4$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 - 16 x - 4 \cdot 12}}{2 \cdot 1}$

Step 1.3.1.5

Multiply $- 4$ by $12$ .

$y = \frac{- 4 \pm \sqrt{16 - 16 x - 48}}{2 \cdot 1}$

Step 1.3.1.6

Subtract $48$ from $16$ .

$y = \frac{- 4 \pm \sqrt{- 16 x - 32}}{2 \cdot 1}$

Step 1.3.1.7

Factor $16$ out of $- 16 x - 32$ .

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Step 1.3.1.7.1

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

$y = \frac{- 4 \pm \sqrt{16 (- x) - 32}}{2 \cdot 1}$

Step 1.3.1.7.2

Factor $16$ out of $- 32$ .

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

Step 1.3.1.7.3

Factor $16$ out of $16 (- x) + 16 (- 2)$ .

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

Step 1.3.1.8

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

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Step 1.3.1.8.1

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

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

Step 1.3.1.8.2

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

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

Step 1.3.1.9

Pull terms out from under the radical.

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

Step 1.3.1.10

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

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

Step 1.3.2

Multiply $2$ by $1$ .

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

Step 1.3.3

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

$y = - 2 \pm 2 \sqrt{- x - 2}$

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 $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (4 x + 12)}}{2 \cdot 1}$

Step 1.4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot (4 x + 12)}}{2 \cdot 1}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 4 (4 x) - 4 \cdot 12}}{2 \cdot 1}$

Step 1.4.1.4

Multiply $4$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 - 16 x - 4 \cdot 12}}{2 \cdot 1}$

Step 1.4.1.5

Multiply $- 4$ by $12$ .

$y = \frac{- 4 \pm \sqrt{16 - 16 x - 48}}{2 \cdot 1}$

Step 1.4.1.6

Subtract $48$ from $16$ .

$y = \frac{- 4 \pm \sqrt{- 16 x - 32}}{2 \cdot 1}$

Step 1.4.1.7

Factor $16$ out of $- 16 x - 32$ .

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Step 1.4.1.7.1

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

$y = \frac{- 4 \pm \sqrt{16 (- x) - 32}}{2 \cdot 1}$

Step 1.4.1.7.2

Factor $16$ out of $- 32$ .

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

Step 1.4.1.7.3

Factor $16$ out of $16 (- x) + 16 (- 2)$ .

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

Step 1.4.1.8

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

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Step 1.4.1.8.1

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

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

Step 1.4.1.8.2

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

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

Step 1.4.1.9

Pull terms out from under the radical.

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

Step 1.4.1.10

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

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

Step 1.4.2

Multiply $2$ by $1$ .

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

Step 1.4.3

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

$y = - 2 \pm 2 \sqrt{- x - 2}$

Step 1.4.4

Change the $\pm$ to $+$ .

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

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 $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (4 x + 12)}}{2 \cdot 1}$

Step 1.5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot (4 x + 12)}}{2 \cdot 1}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 4 (4 x) - 4 \cdot 12}}{2 \cdot 1}$

Step 1.5.1.4

Multiply $4$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 - 16 x - 4 \cdot 12}}{2 \cdot 1}$

Step 1.5.1.5

Multiply $- 4$ by $12$ .

$y = \frac{- 4 \pm \sqrt{16 - 16 x - 48}}{2 \cdot 1}$

Step 1.5.1.6

Subtract $48$ from $16$ .

$y = \frac{- 4 \pm \sqrt{- 16 x - 32}}{2 \cdot 1}$

Step 1.5.1.7

Factor $16$ out of $- 16 x - 32$ .

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Step 1.5.1.7.1

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

$y = \frac{- 4 \pm \sqrt{16 (- x) - 32}}{2 \cdot 1}$

Step 1.5.1.7.2

Factor $16$ out of $- 32$ .

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

Step 1.5.1.7.3

Factor $16$ out of $16 (- x) + 16 (- 2)$ .

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

Step 1.5.1.8

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

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Step 1.5.1.8.1

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

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

Step 1.5.1.8.2

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

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

Step 1.5.1.9

Pull terms out from under the radical.

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

Step 1.5.1.10

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

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

Step 1.5.2

Multiply $2$ by $1$ .

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

Step 1.5.3

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

$y = - 2 \pm 2 \sqrt{- x - 2}$

Step 1.5.4

Change the $\pm$ to $-$ .

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

Step 1.6

The final answer is the combination of both solutions.

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

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

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

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

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

The standard form is $- 2 + 2 \sqrt{- x - 2}, - 2 - 2 \sqrt{- x - 2}$ .

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

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

Step 4