Precalculus Examples

$x^{2} - 3 y^{2} - 8 x + 12 y + 16 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = - 3$ , $b = 12$ , and $c = x^{2} - 8 x + 16$ into the quadratic formula and solve for $y$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (- 3 \cdot (x^{2} - 8 x + 16))}}{2 \cdot - 3}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 3 \cdot (x^{2} - 8 x + 16)}}{2 \cdot - 3}$

Step 3.1.2

Multiply $- 4$ by $- 3$ .

$y = \frac{- 12 \pm \sqrt{144 + 12 \cdot (x^{2} - 8 x + 16)}}{2 \cdot - 3}$

Step 3.1.3

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 + 12 x^{2} + 12 (- 8 x) + 12 \cdot 16}}{2 \cdot - 3}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $- 8$ by $12$ .

$y = \frac{- 12 \pm \sqrt{144 + 12 x^{2} - 96 x + 12 \cdot 16}}{2 \cdot - 3}$

Step 3.1.4.2

Multiply $12$ by $16$ .

$y = \frac{- 12 \pm \sqrt{144 + 12 x^{2} - 96 x + 192}}{2 \cdot - 3}$

Step 3.1.5

Add $144$ and $192$ .

$y = \frac{- 12 \pm \sqrt{12 x^{2} - 96 x + 336}}{2 \cdot - 3}$

Step 3.1.6

Factor $12$ out of $12 x^{2} - 96 x + 336$ .

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Step 3.1.6.1

Factor $12$ out of $12 x^{2}$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2}) - 96 x + 336}}{2 \cdot - 3}$

Step 3.1.6.2

Factor $12$ out of $- 96 x$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2}) + 12 (- 8 x) + 336}}{2 \cdot - 3}$

Step 3.1.6.3

Factor $12$ out of $336$ .

$y = \frac{- 12 \pm \sqrt{12 x^{2} + 12 (- 8 x) + 12 \cdot 28}}{2 \cdot - 3}$

Step 3.1.6.4

Factor $12$ out of $12 x^{2} + 12 (- 8 x)$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2} - 8 x) + 12 \cdot 28}}{2 \cdot - 3}$

Step 3.1.6.5

Factor $12$ out of $12 (x^{2} - 8 x) + 12 \cdot 28$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2} - 8 x + 28)}}{2 \cdot - 3}$

Step 3.1.7

Rewrite $12 (x^{2} - 8 x + 28)$ as $2^{2} \cdot (3 (x^{2} - 8 x + 28))$ .

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Step 3.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 12 \pm \sqrt{4 (3) (x^{2} - 8 x + 28)}}{2 \cdot - 3}$

Step 3.1.7.2

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (x^{2} - 8 x + 28))}}{2 \cdot - 3}$

Step 3.1.7.3

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (x^{2} - 8 x + 28))}}{2 \cdot - 3}$

Step 3.1.8

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{3 (x^{2} - 8 x + 28)}}{2 \cdot - 3}$

Step 3.2

Multiply $2$ by $- 3$ .

$y = \frac{- 12 \pm 2 \sqrt{3 (x^{2} - 8 x + 28)}}{- 6}$

Step 3.3

Simplify $\frac{- 12 \pm 2 \sqrt{3 (x^{2} - 8 x + 28)}}{- 6}$ .

$y = \frac{6 \pm \sqrt{3 (x^{2} - 8 x + 28)}}{3}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 3 \cdot (x^{2} - 8 x + 16)}}{2 \cdot - 3}$

Step 4.1.2

Multiply $- 4$ by $- 3$ .

$y = \frac{- 12 \pm \sqrt{144 + 12 \cdot (x^{2} - 8 x + 16)}}{2 \cdot - 3}$

Step 4.1.3

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 + 12 x^{2} + 12 (- 8 x) + 12 \cdot 16}}{2 \cdot - 3}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $- 8$ by $12$ .

$y = \frac{- 12 \pm \sqrt{144 + 12 x^{2} - 96 x + 12 \cdot 16}}{2 \cdot - 3}$

Step 4.1.4.2

Multiply $12$ by $16$ .

$y = \frac{- 12 \pm \sqrt{144 + 12 x^{2} - 96 x + 192}}{2 \cdot - 3}$

Step 4.1.5

Add $144$ and $192$ .

$y = \frac{- 12 \pm \sqrt{12 x^{2} - 96 x + 336}}{2 \cdot - 3}$

Step 4.1.6

Factor $12$ out of $12 x^{2} - 96 x + 336$ .

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Step 4.1.6.1

Factor $12$ out of $12 x^{2}$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2}) - 96 x + 336}}{2 \cdot - 3}$

Step 4.1.6.2

Factor $12$ out of $- 96 x$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2}) + 12 (- 8 x) + 336}}{2 \cdot - 3}$

Step 4.1.6.3

Factor $12$ out of $336$ .

$y = \frac{- 12 \pm \sqrt{12 x^{2} + 12 (- 8 x) + 12 \cdot 28}}{2 \cdot - 3}$

Step 4.1.6.4

Factor $12$ out of $12 x^{2} + 12 (- 8 x)$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2} - 8 x) + 12 \cdot 28}}{2 \cdot - 3}$

Step 4.1.6.5

Factor $12$ out of $12 (x^{2} - 8 x) + 12 \cdot 28$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2} - 8 x + 28)}}{2 \cdot - 3}$

Step 4.1.7

Rewrite $12 (x^{2} - 8 x + 28)$ as $2^{2} \cdot (3 (x^{2} - 8 x + 28))$ .

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Step 4.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 12 \pm \sqrt{4 (3) (x^{2} - 8 x + 28)}}{2 \cdot - 3}$

Step 4.1.7.2

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (x^{2} - 8 x + 28))}}{2 \cdot - 3}$

Step 4.1.7.3

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (x^{2} - 8 x + 28))}}{2 \cdot - 3}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{3 (x^{2} - 8 x + 28)}}{2 \cdot - 3}$

Step 4.2

Multiply $2$ by $- 3$ .

$y = \frac{- 12 \pm 2 \sqrt{3 (x^{2} - 8 x + 28)}}{- 6}$

Step 4.3

Simplify $\frac{- 12 \pm 2 \sqrt{3 (x^{2} - 8 x + 28)}}{- 6}$ .

$y = \frac{6 \pm \sqrt{3 (x^{2} - 8 x + 28)}}{3}$

Step 4.4

Change the $\pm$ to $+$ .

$y = \frac{6 + \sqrt{3 (x^{2} - 8 x + 28)}}{3}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 3 \cdot (x^{2} - 8 x + 16)}}{2 \cdot - 3}$

Step 5.1.2

Multiply $- 4$ by $- 3$ .

$y = \frac{- 12 \pm \sqrt{144 + 12 \cdot (x^{2} - 8 x + 16)}}{2 \cdot - 3}$

Step 5.1.3

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 + 12 x^{2} + 12 (- 8 x) + 12 \cdot 16}}{2 \cdot - 3}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $- 8$ by $12$ .

$y = \frac{- 12 \pm \sqrt{144 + 12 x^{2} - 96 x + 12 \cdot 16}}{2 \cdot - 3}$

Step 5.1.4.2

Multiply $12$ by $16$ .

$y = \frac{- 12 \pm \sqrt{144 + 12 x^{2} - 96 x + 192}}{2 \cdot - 3}$

Step 5.1.5

Add $144$ and $192$ .

$y = \frac{- 12 \pm \sqrt{12 x^{2} - 96 x + 336}}{2 \cdot - 3}$

Step 5.1.6

Factor $12$ out of $12 x^{2} - 96 x + 336$ .

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Step 5.1.6.1

Factor $12$ out of $12 x^{2}$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2}) - 96 x + 336}}{2 \cdot - 3}$

Step 5.1.6.2

Factor $12$ out of $- 96 x$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2}) + 12 (- 8 x) + 336}}{2 \cdot - 3}$

Step 5.1.6.3

Factor $12$ out of $336$ .

$y = \frac{- 12 \pm \sqrt{12 x^{2} + 12 (- 8 x) + 12 \cdot 28}}{2 \cdot - 3}$

Step 5.1.6.4

Factor $12$ out of $12 x^{2} + 12 (- 8 x)$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2} - 8 x) + 12 \cdot 28}}{2 \cdot - 3}$

Step 5.1.6.5

Factor $12$ out of $12 (x^{2} - 8 x) + 12 \cdot 28$ .

$y = \frac{- 12 \pm \sqrt{12 (x^{2} - 8 x + 28)}}{2 \cdot - 3}$

Step 5.1.7

Rewrite $12 (x^{2} - 8 x + 28)$ as $2^{2} \cdot (3 (x^{2} - 8 x + 28))$ .

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Step 5.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 12 \pm \sqrt{4 (3) (x^{2} - 8 x + 28)}}{2 \cdot - 3}$

Step 5.1.7.2

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (x^{2} - 8 x + 28))}}{2 \cdot - 3}$

Step 5.1.7.3

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (x^{2} - 8 x + 28))}}{2 \cdot - 3}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{3 (x^{2} - 8 x + 28)}}{2 \cdot - 3}$

Step 5.2

Multiply $2$ by $- 3$ .

$y = \frac{- 12 \pm 2 \sqrt{3 (x^{2} - 8 x + 28)}}{- 6}$

Step 5.3

Simplify $\frac{- 12 \pm 2 \sqrt{3 (x^{2} - 8 x + 28)}}{- 6}$ .

$y = \frac{6 \pm \sqrt{3 (x^{2} - 8 x + 28)}}{3}$

Step 5.4

Change the $\pm$ to $-$ .

$y = \frac{6 - \sqrt{3 (x^{2} - 8 x + 28)}}{3}$

Step 6

The final answer is the combination of both solutions.

$y = \frac{6 + \sqrt{3 (x^{2} - 8 x + 28)}}{3}$

$y = \frac{6 - \sqrt{3 (x^{2} - 8 x + 28)}}{3}$

Step 7

Set the radicand in $\sqrt{3 (x^{2} - 8 x + 28)}$ greater than or equal to $0$ to find where the expression is defined.

$3 (x^{2} - 8 x + 28) \geq 0$

Step 8

Solve for $x$ .

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Step 8.1

Divide each term in $3 (x^{2} - 8 x + 28) \geq 0$ by $3$ and simplify.

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Step 8.1.1

Divide each term in $3 (x^{2} - 8 x + 28) \geq 0$ by $3$ .

$\frac{3 (x^{2} - 8 x + 28)}{3} \geq \frac{0}{3}$

Step 8.1.2

Simplify the left side.

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Step 8.1.2.1

Cancel the common factor of $3$ .

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Step 8.1.2.1.1

Cancel the common factor.

$\frac{3 (x^{2} - 8 x + 28)}{3} \geq \frac{0}{3}$

Step 8.1.2.1.2

Divide $x^{2} - 8 x + 28$ by $1$ .

$x^{2} - 8 x + 28 \geq \frac{0}{3}$

Step 8.1.3

Simplify the right side.

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Step 8.1.3.1

Divide $0$ by $3$ .

$x^{2} - 8 x + 28 \geq 0$

Step 8.2

Convert the inequality to an equation.

$x^{2} - 8 x + 28 = 0$

Step 8.3

Use the quadratic formula to find the solutions.

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

Step 8.4

Substitute the values $a = 1$ , $b = - 8$ , and $c = 28$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 28)}}{2 \cdot 1}$

Step 8.5

Simplify.

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Step 8.5.1

Simplify the numerator.

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Step 8.5.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 28}}{2 \cdot 1}$

Step 8.5.1.2

Multiply $- 4 \cdot 1 \cdot 28$ .

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Step 8.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 28}}{2 \cdot 1}$

Step 8.5.1.2.2

Multiply $- 4$ by $28$ .

$x = \frac{8 \pm \sqrt{64 - 112}}{2 \cdot 1}$

Step 8.5.1.3

Subtract $112$ from $64$ .

$x = \frac{8 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 8.5.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 8.5.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 8.5.1.6

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

$x = \frac{8 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 8.5.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 8.5.1.7.1

Factor $16$ out of $48$ .

$x = \frac{8 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 8.5.1.7.2

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

$x = \frac{8 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 8.5.1.8

Pull terms out from under the radical.

$x = \frac{8 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 8.5.1.9

Move $4$ to the left of $i$ .

$x = \frac{8 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 8.5.2

Multiply $2$ by $1$ .

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

Step 8.5.3

Simplify $\frac{8 \pm 4 i \sqrt{3}}{2}$ .

$x = 4 \pm 2 i \sqrt{3}$

Step 8.6

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

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Step 8.6.1

Simplify the numerator.

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Step 8.6.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 28}}{2 \cdot 1}$

Step 8.6.1.2

Multiply $- 4 \cdot 1 \cdot 28$ .

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Step 8.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 28}}{2 \cdot 1}$

Step 8.6.1.2.2

Multiply $- 4$ by $28$ .

$x = \frac{8 \pm \sqrt{64 - 112}}{2 \cdot 1}$

Step 8.6.1.3

Subtract $112$ from $64$ .

$x = \frac{8 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 8.6.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 8.6.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 8.6.1.6

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

$x = \frac{8 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 8.6.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 8.6.1.7.1

Factor $16$ out of $48$ .

$x = \frac{8 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 8.6.1.7.2

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

$x = \frac{8 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 8.6.1.8

Pull terms out from under the radical.

$x = \frac{8 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 8.6.1.9

Move $4$ to the left of $i$ .

$x = \frac{8 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 8.6.2

Multiply $2$ by $1$ .

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

Step 8.6.3

Simplify $\frac{8 \pm 4 i \sqrt{3}}{2}$ .

$x = 4 \pm 2 i \sqrt{3}$

Step 8.6.4

Change the $\pm$ to $+$ .

$x = 4 + 2 i \sqrt{3}$

Step 8.7

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

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Step 8.7.1

Simplify the numerator.

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Step 8.7.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 28}}{2 \cdot 1}$

Step 8.7.1.2

Multiply $- 4 \cdot 1 \cdot 28$ .

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Step 8.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 28}}{2 \cdot 1}$

Step 8.7.1.2.2

Multiply $- 4$ by $28$ .

$x = \frac{8 \pm \sqrt{64 - 112}}{2 \cdot 1}$

Step 8.7.1.3

Subtract $112$ from $64$ .

$x = \frac{8 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 8.7.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 8.7.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 8.7.1.6

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

$x = \frac{8 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 8.7.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 8.7.1.7.1

Factor $16$ out of $48$ .

$x = \frac{8 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 8.7.1.7.2

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

$x = \frac{8 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 8.7.1.8

Pull terms out from under the radical.

$x = \frac{8 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 8.7.1.9

Move $4$ to the left of $i$ .

$x = \frac{8 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 8.7.2

Multiply $2$ by $1$ .

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

Step 8.7.3

Simplify $\frac{8 \pm 4 i \sqrt{3}}{2}$ .

$x = 4 \pm 2 i \sqrt{3}$

Step 8.7.4

Change the $\pm$ to $-$ .

$x = 4 - 2 i \sqrt{3}$

Step 8.8

Identify the leading coefficient.

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Step 8.8.1

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

$x^{2}$

Step 8.8.2

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

$1$

Step 8.9

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $x^{2} - 8 x + 28$ is always greater than $0$ .

All real numbers

Step 9

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 10

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

Interval Notation:

$(- \infty, 0] \cup [4, \infty)$

Set-Builder Notation:

${y | y \leq 0, y \geq 4}$

Step 11

Determine the domain and range.

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

Range: $(- \infty, 0] \cup [4, \infty), {y | y \leq 0, y \geq 4}$

Step 12