Calculus Examples

$f (x) = 2 x^{2} - 2 x y + 2 y^{2} + 4$

Step 1

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

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

Step 2

Subtract $x$ from both sides of the equation.

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

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Add parentheses.

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

Step 5.1.2

Let $u = 2 \cdot (2 x^{2} + 4 - x)$ . Substitute $u$ for all occurrences of $2 \cdot (2 x^{2} + 4 - x)$ .

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Step 5.1.2.1

Apply the product rule to $- 2 x$ .

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

Step 5.1.2.2

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

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

Step 5.1.3

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

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Step 5.1.3.1

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

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

Step 5.1.3.2

Factor $4$ out of $- 4 u$ .

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

Step 5.1.3.3

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

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

Step 5.1.4

Replace all occurrences of $u$ with $2 \cdot (2 x^{2} + 4 - x)$ .

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

Step 5.1.5

Simplify.

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Step 5.1.5.1

Simplify each term.

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Step 5.1.5.1.1

Apply the distributive property.

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

Step 5.1.5.1.2

Simplify.

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Step 5.1.5.1.2.1

Multiply $2$ by $2$ .

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

Step 5.1.5.1.2.2

Multiply $2$ by $4$ .

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

Step 5.1.5.1.2.3

Multiply $- 1$ by $2$ .

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

Step 5.1.5.1.3

Apply the distributive property.

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

Step 5.1.5.1.4

Simplify.

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Step 5.1.5.1.4.1

Multiply $4$ by $- 1$ .

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

Step 5.1.5.1.4.2

Multiply $- 1$ by $8$ .

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

Step 5.1.5.1.4.3

Multiply $- 2$ by $- 1$ .

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

Step 5.1.5.2

Subtract $4 x^{2}$ from $x^{2}$ .

$y = \frac{2 x \pm \sqrt{4 (- 3 x^{2} - 8 + 2 x)}}{2 \cdot 2}$

Step 5.1.6

Reorder terms.

$y = \frac{2 x \pm \sqrt{4 (- 3 x^{2} + 2 x - 8)}}{2 \cdot 2}$

Step 5.1.7

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

$y = \frac{2 x \pm \sqrt{2^{2} (- 3 x^{2} + 2 x - 8)}}{2 \cdot 2}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{2 x \pm 2 \sqrt{- 3 x^{2} + 2 x - 8}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$y = \frac{2 x \pm 2 \sqrt{- 3 x^{2} + 2 x - 8}}{4}$

Step 5.3

Simplify $\frac{2 x \pm 2 \sqrt{- 3 x^{2} + 2 x - 8}}{4}$ .

$y = \frac{x \pm \sqrt{- 3 x^{2} + 2 x - 8}}{2}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Add parentheses.

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

Step 6.1.2

Let $u = 2 \cdot (2 x^{2} + 4 - x)$ . Substitute $u$ for all occurrences of $2 \cdot (2 x^{2} + 4 - x)$ .

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Step 6.1.2.1

Apply the product rule to $- 2 x$ .

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

Step 6.1.2.2

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

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

Step 6.1.3

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

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Step 6.1.3.1

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

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

Step 6.1.3.2

Factor $4$ out of $- 4 u$ .

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

Step 6.1.3.3

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

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

Step 6.1.4

Replace all occurrences of $u$ with $2 \cdot (2 x^{2} + 4 - x)$ .

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

Step 6.1.5

Simplify.

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Step 6.1.5.1

Simplify each term.

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Step 6.1.5.1.1

Apply the distributive property.

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

Step 6.1.5.1.2

Simplify.

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Step 6.1.5.1.2.1

Multiply $2$ by $2$ .

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

Step 6.1.5.1.2.2

Multiply $2$ by $4$ .

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

Step 6.1.5.1.2.3

Multiply $- 1$ by $2$ .

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

Step 6.1.5.1.3

Apply the distributive property.

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

Step 6.1.5.1.4

Simplify.

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Step 6.1.5.1.4.1

Multiply $4$ by $- 1$ .

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

Step 6.1.5.1.4.2

Multiply $- 1$ by $8$ .

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

Step 6.1.5.1.4.3

Multiply $- 2$ by $- 1$ .

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

Step 6.1.5.2

Subtract $4 x^{2}$ from $x^{2}$ .

$y = \frac{2 x \pm \sqrt{4 (- 3 x^{2} - 8 + 2 x)}}{2 \cdot 2}$

Step 6.1.6

Reorder terms.

$y = \frac{2 x \pm \sqrt{4 (- 3 x^{2} + 2 x - 8)}}{2 \cdot 2}$

Step 6.1.7

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

$y = \frac{2 x \pm \sqrt{2^{2} (- 3 x^{2} + 2 x - 8)}}{2 \cdot 2}$

Step 6.1.8

Pull terms out from under the radical.

$y = \frac{2 x \pm 2 \sqrt{- 3 x^{2} + 2 x - 8}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$y = \frac{2 x \pm 2 \sqrt{- 3 x^{2} + 2 x - 8}}{4}$

Step 6.3

Simplify $\frac{2 x \pm 2 \sqrt{- 3 x^{2} + 2 x - 8}}{4}$ .

$y = \frac{x \pm \sqrt{- 3 x^{2} + 2 x - 8}}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$y = \frac{x + \sqrt{- 3 x^{2} + 2 x - 8}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Add parentheses.

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

Step 7.1.2

Let $u = 2 \cdot (2 x^{2} + 4 - x)$ . Substitute $u$ for all occurrences of $2 \cdot (2 x^{2} + 4 - x)$ .

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Step 7.1.2.1

Apply the product rule to $- 2 x$ .

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

Step 7.1.2.2

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

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

Step 7.1.3

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

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Step 7.1.3.1

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

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

Step 7.1.3.2

Factor $4$ out of $- 4 u$ .

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

Step 7.1.3.3

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

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

Step 7.1.4

Replace all occurrences of $u$ with $2 \cdot (2 x^{2} + 4 - x)$ .

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

Step 7.1.5

Simplify.

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Step 7.1.5.1

Simplify each term.

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Step 7.1.5.1.1

Apply the distributive property.

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

Step 7.1.5.1.2

Simplify.

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Step 7.1.5.1.2.1

Multiply $2$ by $2$ .

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

Step 7.1.5.1.2.2

Multiply $2$ by $4$ .

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

Step 7.1.5.1.2.3

Multiply $- 1$ by $2$ .

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

Step 7.1.5.1.3

Apply the distributive property.

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

Step 7.1.5.1.4

Simplify.

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Step 7.1.5.1.4.1

Multiply $4$ by $- 1$ .

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

Step 7.1.5.1.4.2

Multiply $- 1$ by $8$ .

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

Step 7.1.5.1.4.3

Multiply $- 2$ by $- 1$ .

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

Step 7.1.5.2

Subtract $4 x^{2}$ from $x^{2}$ .

$y = \frac{2 x \pm \sqrt{4 (- 3 x^{2} - 8 + 2 x)}}{2 \cdot 2}$

Step 7.1.6

Reorder terms.

$y = \frac{2 x \pm \sqrt{4 (- 3 x^{2} + 2 x - 8)}}{2 \cdot 2}$

Step 7.1.7

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

$y = \frac{2 x \pm \sqrt{2^{2} (- 3 x^{2} + 2 x - 8)}}{2 \cdot 2}$

Step 7.1.8

Pull terms out from under the radical.

$y = \frac{2 x \pm 2 \sqrt{- 3 x^{2} + 2 x - 8}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$y = \frac{2 x \pm 2 \sqrt{- 3 x^{2} + 2 x - 8}}{4}$

Step 7.3

Simplify $\frac{2 x \pm 2 \sqrt{- 3 x^{2} + 2 x - 8}}{4}$ .

$y = \frac{x \pm \sqrt{- 3 x^{2} + 2 x - 8}}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$y = \frac{x - \sqrt{- 3 x^{2} + 2 x - 8}}{2}$

Step 8

The final answer is the combination of both solutions.

$y = \frac{x + \sqrt{- 3 x^{2} + 2 x - 8}}{2}$

$y = \frac{x - \sqrt{- 3 x^{2} + 2 x - 8}}{2}$

Step 9

Set the radicand in $\sqrt{- 3 x^{2} + 2 x - 8}$ greater than or equal to $0$ to find where the expression is defined.

$- 3 x^{2} + 2 x - 8 \geq 0$

Step 10

Solve for $x$ .

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Step 10.1

Convert the inequality to an equation.

$- 3 x^{2} + 2 x - 8 = 0$

Step 10.2

Use the quadratic formula to find the solutions.

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

Step 10.3

Substitute the values $a = - 3$ , $b = 2$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (- 3 \cdot - 8)}}{2 \cdot - 3}$

Step 10.4

Simplify.

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Step 10.4.1

Simplify the numerator.

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Step 10.4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 3 \cdot - 8}}{2 \cdot - 3}$

Step 10.4.1.2

Multiply $- 4 \cdot - 3 \cdot - 8$ .

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Step 10.4.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 2 \pm \sqrt{4 + 12 \cdot - 8}}{2 \cdot - 3}$

Step 10.4.1.2.2

Multiply $12$ by $- 8$ .

$x = \frac{- 2 \pm \sqrt{4 - 96}}{2 \cdot - 3}$

Step 10.4.1.3

Subtract $96$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 92}}{2 \cdot - 3}$

Step 10.4.1.4

Rewrite $- 92$ as $- 1 (92)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 92}}{2 \cdot - 3}$

Step 10.4.1.5

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot - 3}$

Step 10.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{92}}{2 \cdot - 3}$

Step 10.4.1.7

Rewrite $92$ as $2^{2} \cdot 23$ .

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Step 10.4.1.7.1

Factor $4$ out of $92$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (23)}}{2 \cdot - 3}$

Step 10.4.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot - 3}$

Step 10.4.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{23})}{2 \cdot - 3}$

Step 10.4.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{23}}{2 \cdot - 3}$

Step 10.4.2

Multiply $2$ by $- 3$ .

$x = \frac{- 2 \pm 2 i \sqrt{23}}{- 6}$

Step 10.4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{23}}{- 6}$ .

$x = \frac{1 \pm i \sqrt{23}}{3}$

Step 10.5

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

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Step 10.5.1

Simplify the numerator.

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Step 10.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 3 \cdot - 8}}{2 \cdot - 3}$

Step 10.5.1.2

Multiply $- 4 \cdot - 3 \cdot - 8$ .

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Step 10.5.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 2 \pm \sqrt{4 + 12 \cdot - 8}}{2 \cdot - 3}$

Step 10.5.1.2.2

Multiply $12$ by $- 8$ .

$x = \frac{- 2 \pm \sqrt{4 - 96}}{2 \cdot - 3}$

Step 10.5.1.3

Subtract $96$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 92}}{2 \cdot - 3}$

Step 10.5.1.4

Rewrite $- 92$ as $- 1 (92)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 92}}{2 \cdot - 3}$

Step 10.5.1.5

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot - 3}$

Step 10.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{92}}{2 \cdot - 3}$

Step 10.5.1.7

Rewrite $92$ as $2^{2} \cdot 23$ .

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Step 10.5.1.7.1

Factor $4$ out of $92$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (23)}}{2 \cdot - 3}$

Step 10.5.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot - 3}$

Step 10.5.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{23})}{2 \cdot - 3}$

Step 10.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{23}}{2 \cdot - 3}$

Step 10.5.2

Multiply $2$ by $- 3$ .

$x = \frac{- 2 \pm 2 i \sqrt{23}}{- 6}$

Step 10.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{23}}{- 6}$ .

$x = \frac{1 \pm i \sqrt{23}}{3}$

Step 10.5.4

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{23}}{3}$

Step 10.6

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

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Step 10.6.1

Simplify the numerator.

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Step 10.6.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 3 \cdot - 8}}{2 \cdot - 3}$

Step 10.6.1.2

Multiply $- 4 \cdot - 3 \cdot - 8$ .

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Step 10.6.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 2 \pm \sqrt{4 + 12 \cdot - 8}}{2 \cdot - 3}$

Step 10.6.1.2.2

Multiply $12$ by $- 8$ .

$x = \frac{- 2 \pm \sqrt{4 - 96}}{2 \cdot - 3}$

Step 10.6.1.3

Subtract $96$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 92}}{2 \cdot - 3}$

Step 10.6.1.4

Rewrite $- 92$ as $- 1 (92)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 92}}{2 \cdot - 3}$

Step 10.6.1.5

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot - 3}$

Step 10.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{92}}{2 \cdot - 3}$

Step 10.6.1.7

Rewrite $92$ as $2^{2} \cdot 23$ .

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Step 10.6.1.7.1

Factor $4$ out of $92$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (23)}}{2 \cdot - 3}$

Step 10.6.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot - 3}$

Step 10.6.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{23})}{2 \cdot - 3}$

Step 10.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{23}}{2 \cdot - 3}$

Step 10.6.2

Multiply $2$ by $- 3$ .

$x = \frac{- 2 \pm 2 i \sqrt{23}}{- 6}$

Step 10.6.3

Simplify $\frac{- 2 \pm 2 i \sqrt{23}}{- 6}$ .

$x = \frac{1 \pm i \sqrt{23}}{3}$

Step 10.6.4

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{23}}{3}$

Step 10.7

Identify the leading coefficient.

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Step 10.7.1

The leading term in a polynomial is the term with the highest degree.

$- 3 x^{2}$

Step 10.7.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 3$

Step 10.8

Since there are no real x-intercepts and the leading coefficient is negative, the parabola opens down and $- 3 x^{2} + 2 x - 8$ is always less than $0$ .

No solution

Step 11

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 12

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 13

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 14