Precalculus Examples

$x^{2} + y^{2} - 6 x - 4 y - 12 = 0$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$x^{2} + {(0)}^{2} - 6 x - 4 \cdot 0 - 12 = 0$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $x^{2} + {(0)}^{2} - 6 x - 4 \cdot 0 - 12$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$x^{2} + 0 - 6 x - 4 \cdot 0 - 12 = 0$

Step 1.2.1.1.2

Multiply $- 4$ by $0$ .

$x^{2} + 0 - 6 x + 0 - 12 = 0$

Step 1.2.1.2

Combine the opposite terms in $x^{2} + 0 - 6 x + 0 - 12$ .

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Step 1.2.1.2.1

Add $x^{2}$ and $0$ .

$x^{2} - 6 x + 0 - 12 = 0$

Step 1.2.1.2.2

Add $x^{2} - 6 x$ and $0$ .

$x^{2} - 6 x - 12 = 0$

Step 1.2.2

Use the quadratic formula to find the solutions.

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

Step 1.2.3

Substitute the values $a = 1$ , $b = - 6$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 12)}}{2 \cdot 1}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 1.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 12$ .

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Step 1.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 12}}{2 \cdot 1}$

Step 1.2.4.1.2.2

Multiply $- 4$ by $- 12$ .

$x = \frac{6 \pm \sqrt{36 + 48}}{2 \cdot 1}$

Step 1.2.4.1.3

Add $36$ and $48$ .

$x = \frac{6 \pm \sqrt{84}}{2 \cdot 1}$

Step 1.2.4.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 1.2.4.1.4.1

Factor $4$ out of $84$ .

$x = \frac{6 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 1.2.4.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 1.2.4.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 1.2.4.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{21}}{2}$

Step 1.2.4.3

Simplify $\frac{6 \pm 2 \sqrt{21}}{2}$ .

$x = 3 \pm \sqrt{21}$

Step 1.2.5

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

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 12$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 12}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply $- 4$ by $- 12$ .

$x = \frac{6 \pm \sqrt{36 + 48}}{2 \cdot 1}$

Step 1.2.5.1.3

Add $36$ and $48$ .

$x = \frac{6 \pm \sqrt{84}}{2 \cdot 1}$

Step 1.2.5.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 1.2.5.1.4.1

Factor $4$ out of $84$ .

$x = \frac{6 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 1.2.5.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 1.2.5.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 1.2.5.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{21}}{2}$

Step 1.2.5.3

Simplify $\frac{6 \pm 2 \sqrt{21}}{2}$ .

$x = 3 \pm \sqrt{21}$

Step 1.2.5.4

Change the $\pm$ to $+$ .

$x = 3 + \sqrt{21}$

Step 1.2.6

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

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 12$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 12}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $- 12$ .

$x = \frac{6 \pm \sqrt{36 + 48}}{2 \cdot 1}$

Step 1.2.6.1.3

Add $36$ and $48$ .

$x = \frac{6 \pm \sqrt{84}}{2 \cdot 1}$

Step 1.2.6.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 1.2.6.1.4.1

Factor $4$ out of $84$ .

$x = \frac{6 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 1.2.6.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{21}}{2}$

Step 1.2.6.3

Simplify $\frac{6 \pm 2 \sqrt{21}}{2}$ .

$x = 3 \pm \sqrt{21}$

Step 1.2.6.4

Change the $\pm$ to $-$ .

$x = 3 - \sqrt{21}$

Step 1.2.7

The final answer is the combination of both solutions.

$x = 3 + \sqrt{21}, 3 - \sqrt{21}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3 + \sqrt{21}, 0), (3 - \sqrt{21}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

${(0)}^{2} + y^{2} - 6 \cdot 0 - 4 y - 12 = 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify ${(0)}^{2} + y^{2} - 6 \cdot 0 - 4 y - 12$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$0 + y^{2} - 6 \cdot 0 - 4 y - 12 = 0$

Step 2.2.1.1.2

Multiply $- 6$ by $0$ .

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

Step 2.2.1.2

Combine the opposite terms in $0 + y^{2} + 0 - 4 y - 12$ .

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Step 2.2.1.2.1

Add $0$ and $y^{2}$ .

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

Step 2.2.1.2.2

Add $y^{2}$ and $0$ .

$y^{2} - 4 y - 12 = 0$

Step 2.2.2

Factor $y^{2} - 4 y - 12$ using the AC method.

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Step 2.2.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 2.2.2.2

Write the factored form using these integers.

$(y - 6) (y + 2) = 0$

Step 2.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 6 = 0$

$y + 2 = 0$

Step 2.2.4

Set $y - 6$ equal to $0$ and solve for $y$ .

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Step 2.2.4.1

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Step 2.2.4.2

Add $6$ to both sides of the equation.

$y = 6$

Step 2.2.5

Set $y + 2$ equal to $0$ and solve for $y$ .

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Step 2.2.5.1

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

Step 2.2.5.2

Subtract $2$ from both sides of the equation.

$y = - 2$

Step 2.2.6

The final solution is all the values that make $(y - 6) (y + 2) = 0$ true.

$y = 6, - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 6), (0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(3 + \sqrt{21}, 0), (3 - \sqrt{21}, 0)$

y-intercept(s): $(0, 6), (0, - 2)$

Step 4