Trigonometry Examples

$x^{2} + y^{2} + 4 x - 8 y + 24 = 0$

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 $a = 1$ , $b = - 8$ , and $c = x^{2} + 4 x + 24$ into the quadratic formula and solve for $y$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot (x^{2} + 4 x + 24))}}{2 \cdot 1}$

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 $- 8$ to the power of $2$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot (x^{2} + 4 x + 24)}}{2 \cdot 1}$

Step 1.3.1.2

Multiply $- 4$ by $1$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (x^{2} + 4 x + 24)}}{2 \cdot 1}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{8 \pm \sqrt{64 - 4 x^{2} - 4 (4 x) - 4 \cdot 24}}{2 \cdot 1}$

Step 1.3.1.4

Simplify.

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Step 1.3.1.4.1

Multiply $4$ by $- 4$ .

$y = \frac{8 \pm \sqrt{64 - 4 x^{2} - 16 x - 4 \cdot 24}}{2 \cdot 1}$

Step 1.3.1.4.2

Multiply $- 4$ by $24$ .

$y = \frac{8 \pm \sqrt{64 - 4 x^{2} - 16 x - 96}}{2 \cdot 1}$

Step 1.3.1.5

Subtract $96$ from $64$ .

$y = \frac{8 \pm \sqrt{- 4 x^{2} - 16 x - 32}}{2 \cdot 1}$

Step 1.3.1.6

Factor $4$ out of $- 4 x^{2} - 16 x - 32$ .

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Step 1.3.1.6.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2}) - 16 x - 32}}{2 \cdot 1}$

Step 1.3.1.6.2

Factor $4$ out of $- 16 x$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) - 32}}{2 \cdot 1}$

Step 1.3.1.6.3

Factor $4$ out of $- 32$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 4 (- 8)}}{2 \cdot 1}$

Step 1.3.1.6.4

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

$y = \frac{8 \pm \sqrt{4 (- x^{2} - 4 x) + 4 (- 8)}}{2 \cdot 1}$

Step 1.3.1.6.5

Factor $4$ out of $4 (- x^{2} - 4 x) + 4 (- 8)$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2} - 4 x - 8)}}{2 \cdot 1}$

Step 1.3.1.7

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

$y = \frac{8 \pm \sqrt{2^{2} (- x^{2} - 4 x - 8)}}{2 \cdot 1}$

Step 1.3.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm 2 \sqrt{- x^{2} - 4 x - 8}}{2 \cdot 1}$

Step 1.3.2

Multiply $2$ by $1$ .

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

Step 1.3.3

Simplify $\frac{8 \pm 2 \sqrt{- x^{2} - 4 x - 8}}{2}$ .

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

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 $- 8$ to the power of $2$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot (x^{2} + 4 x + 24)}}{2 \cdot 1}$

Step 1.4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (x^{2} + 4 x + 24)}}{2 \cdot 1}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{8 \pm \sqrt{64 - 4 x^{2} - 4 (4 x) - 4 \cdot 24}}{2 \cdot 1}$

Step 1.4.1.4

Simplify.

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Step 1.4.1.4.1

Multiply $4$ by $- 4$ .

$y = \frac{8 \pm \sqrt{64 - 4 x^{2} - 16 x - 4 \cdot 24}}{2 \cdot 1}$

Step 1.4.1.4.2

Multiply $- 4$ by $24$ .

$y = \frac{8 \pm \sqrt{64 - 4 x^{2} - 16 x - 96}}{2 \cdot 1}$

Step 1.4.1.5

Subtract $96$ from $64$ .

$y = \frac{8 \pm \sqrt{- 4 x^{2} - 16 x - 32}}{2 \cdot 1}$

Step 1.4.1.6

Factor $4$ out of $- 4 x^{2} - 16 x - 32$ .

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Step 1.4.1.6.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2}) - 16 x - 32}}{2 \cdot 1}$

Step 1.4.1.6.2

Factor $4$ out of $- 16 x$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) - 32}}{2 \cdot 1}$

Step 1.4.1.6.3

Factor $4$ out of $- 32$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 4 (- 8)}}{2 \cdot 1}$

Step 1.4.1.6.4

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

$y = \frac{8 \pm \sqrt{4 (- x^{2} - 4 x) + 4 (- 8)}}{2 \cdot 1}$

Step 1.4.1.6.5

Factor $4$ out of $4 (- x^{2} - 4 x) + 4 (- 8)$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2} - 4 x - 8)}}{2 \cdot 1}$

Step 1.4.1.7

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

$y = \frac{8 \pm \sqrt{2^{2} (- x^{2} - 4 x - 8)}}{2 \cdot 1}$

Step 1.4.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm 2 \sqrt{- x^{2} - 4 x - 8}}{2 \cdot 1}$

Step 1.4.2

Multiply $2$ by $1$ .

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

Step 1.4.3

Simplify $\frac{8 \pm 2 \sqrt{- x^{2} - 4 x - 8}}{2}$ .

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

Step 1.4.4

Change the $\pm$ to $+$ .

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

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 $- 8$ to the power of $2$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot (x^{2} + 4 x + 24)}}{2 \cdot 1}$

Step 1.5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (x^{2} + 4 x + 24)}}{2 \cdot 1}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{8 \pm \sqrt{64 - 4 x^{2} - 4 (4 x) - 4 \cdot 24}}{2 \cdot 1}$

Step 1.5.1.4

Simplify.

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Step 1.5.1.4.1

Multiply $4$ by $- 4$ .

$y = \frac{8 \pm \sqrt{64 - 4 x^{2} - 16 x - 4 \cdot 24}}{2 \cdot 1}$

Step 1.5.1.4.2

Multiply $- 4$ by $24$ .

$y = \frac{8 \pm \sqrt{64 - 4 x^{2} - 16 x - 96}}{2 \cdot 1}$

Step 1.5.1.5

Subtract $96$ from $64$ .

$y = \frac{8 \pm \sqrt{- 4 x^{2} - 16 x - 32}}{2 \cdot 1}$

Step 1.5.1.6

Factor $4$ out of $- 4 x^{2} - 16 x - 32$ .

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Step 1.5.1.6.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2}) - 16 x - 32}}{2 \cdot 1}$

Step 1.5.1.6.2

Factor $4$ out of $- 16 x$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) - 32}}{2 \cdot 1}$

Step 1.5.1.6.3

Factor $4$ out of $- 32$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 4 (- 8)}}{2 \cdot 1}$

Step 1.5.1.6.4

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

$y = \frac{8 \pm \sqrt{4 (- x^{2} - 4 x) + 4 (- 8)}}{2 \cdot 1}$

Step 1.5.1.6.5

Factor $4$ out of $4 (- x^{2} - 4 x) + 4 (- 8)$ .

$y = \frac{8 \pm \sqrt{4 (- x^{2} - 4 x - 8)}}{2 \cdot 1}$

Step 1.5.1.7

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

$y = \frac{8 \pm \sqrt{2^{2} (- x^{2} - 4 x - 8)}}{2 \cdot 1}$

Step 1.5.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm 2 \sqrt{- x^{2} - 4 x - 8}}{2 \cdot 1}$

Step 1.5.2

Multiply $2$ by $1$ .

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

Step 1.5.3

Simplify $\frac{8 \pm 2 \sqrt{- x^{2} - 4 x - 8}}{2}$ .

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

Step 1.5.4

Change the $\pm$ to $-$ .

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

Step 1.6

The final answer is the combination of both solutions.

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

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

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

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

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.