Precalculus Examples

$3 x^{2} + 3 y^{2} - 24 x + 12 y - 15 = 0$

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 $x$ .

$3 x^{2} + 3 {(0)}^{2} - 24 x + 12 (0) - 15 = 0$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $3 x^{2} + 3 {(0)}^{2} - 24 x + 12 (0) - 15$ .

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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 $0$ .

$3 x^{2} + 3 \cdot 0 - 24 x + 12 (0) - 15 = 0$

Step 1.2.1.1.2

Multiply $3$ by $0$ .

$3 x^{2} + 0 - 24 x + 12 (0) - 15 = 0$

Step 1.2.1.1.3

Multiply $12$ by $0$ .

$3 x^{2} + 0 - 24 x + 0 - 15 = 0$

Step 1.2.1.2

Combine the opposite terms in $3 x^{2} + 0 - 24 x + 0 - 15$ .

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Step 1.2.1.2.1

Add $3 x^{2}$ and $0$ .

$3 x^{2} - 24 x + 0 - 15 = 0$

Step 1.2.1.2.2

Add $3 x^{2} - 24 x$ and $0$ .

$3 x^{2} - 24 x - 15 = 0$

Step 1.2.2

Factor $3$ out of $3 x^{2} - 24 x - 15$ .

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Step 1.2.2.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) - 24 x - 15 = 0$

Step 1.2.2.2

Factor $3$ out of $- 24 x$ .

$3 (x^{2}) + 3 (- 8 x) - 15 = 0$

Step 1.2.2.3

Factor $3$ out of $- 15$ .

$3 x^{2} + 3 (- 8 x) + 3 \cdot - 5 = 0$

Step 1.2.2.4

Factor $3$ out of $3 x^{2} + 3 (- 8 x)$ .

$3 (x^{2} - 8 x) + 3 \cdot - 5 = 0$

Step 1.2.2.5

Factor $3$ out of $3 (x^{2} - 8 x) + 3 \cdot - 5$ .

$3 (x^{2} - 8 x - 5) = 0$

Step 1.2.3

Divide each term in $3 (x^{2} - 8 x - 5) = 0$ by $3$ and simplify.

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Step 1.2.3.1

Divide each term in $3 (x^{2} - 8 x - 5) = 0$ by $3$ .

$\frac{3 (x^{2} - 8 x - 5)}{3} = \frac{0}{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{3 (x^{2} - 8 x - 5)}{3} = \frac{0}{3}$

Step 1.2.3.2.1.2

Divide $x^{2} - 8 x - 5$ by $1$ .

$x^{2} - 8 x - 5 = \frac{0}{3}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $3$ .

$x^{2} - 8 x - 5 = 0$

Step 1.2.4

Use the quadratic formula to find the solutions.

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

Step 1.2.5

Substitute the values $a = 1$ , $b = - 8$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 5}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{8 \pm \sqrt{64 + 20}}{2 \cdot 1}$

Step 1.2.6.1.3

Add $64$ and $20$ .

$x = \frac{8 \pm \sqrt{84}}{2 \cdot 1}$

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 $84$ .

$x = \frac{8 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 1.2.6.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{21}}{2}$

Step 1.2.6.3

Simplify $\frac{8 \pm 2 \sqrt{21}}{2}$ .

$x = 4 \pm \sqrt{21}$

Step 1.2.7

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

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 5}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{8 \pm \sqrt{64 + 20}}{2 \cdot 1}$

Step 1.2.7.1.3

Add $64$ and $20$ .

$x = \frac{8 \pm \sqrt{84}}{2 \cdot 1}$

Step 1.2.7.1.4

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

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Step 1.2.7.1.4.1

Factor $4$ out of $84$ .

$x = \frac{8 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 1.2.7.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{21}}{2}$

Step 1.2.7.3

Simplify $\frac{8 \pm 2 \sqrt{21}}{2}$ .

$x = 4 \pm \sqrt{21}$

Step 1.2.7.4

Change the $\pm$ to $+$ .

$x = 4 + \sqrt{21}$

Step 1.2.8

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

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Step 1.2.8.1

Simplify the numerator.

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Step 1.2.8.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 1.2.8.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 5}}{2 \cdot 1}$

Step 1.2.8.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{8 \pm \sqrt{64 + 20}}{2 \cdot 1}$

Step 1.2.8.1.3

Add $64$ and $20$ .

$x = \frac{8 \pm \sqrt{84}}{2 \cdot 1}$

Step 1.2.8.1.4

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

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Step 1.2.8.1.4.1

Factor $4$ out of $84$ .

$x = \frac{8 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 1.2.8.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 1.2.8.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 1.2.8.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{21}}{2}$

Step 1.2.8.3

Simplify $\frac{8 \pm 2 \sqrt{21}}{2}$ .

$x = 4 \pm \sqrt{21}$

Step 1.2.8.4

Change the $\pm$ to $-$ .

$x = 4 - \sqrt{21}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = 4 + \sqrt{21}, 4 - \sqrt{21}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(4 + \sqrt{21}, 0), (4 - \sqrt{21}, 0)$

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 $y$ .

$3 {(0)}^{2} + 3 y^{2} - 24 \cdot 0 + 12 y - 15 = 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $3 {(0)}^{2} + 3 y^{2} - 24 \cdot 0 + 12 y - 15$ .

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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 $0$ .

$3 \cdot 0 + 3 y^{2} - 24 \cdot 0 + 12 y - 15 = 0$

Step 2.2.1.1.2

Multiply $3$ by $0$ .

$0 + 3 y^{2} - 24 \cdot 0 + 12 y - 15 = 0$

Step 2.2.1.1.3

Multiply $- 24$ by $0$ .

$0 + 3 y^{2} + 0 + 12 y - 15 = 0$

Step 2.2.1.2

Combine the opposite terms in $0 + 3 y^{2} + 0 + 12 y - 15$ .

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Step 2.2.1.2.1

Add $0$ and $3 y^{2}$ .

$3 y^{2} + 0 + 12 y - 15 = 0$

Step 2.2.1.2.2

Add $3 y^{2}$ and $0$ .

$3 y^{2} + 12 y - 15 = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $3$ out of $3 y^{2} + 12 y - 15$ .

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Step 2.2.2.1.1

Factor $3$ out of $3 y^{2}$ .

$3 (y^{2}) + 12 y - 15 = 0$

Step 2.2.2.1.2

Factor $3$ out of $12 y$ .

$3 (y^{2}) + 3 (4 y) - 15 = 0$

Step 2.2.2.1.3

Factor $3$ out of $- 15$ .

$3 y^{2} + 3 (4 y) + 3 \cdot - 5 = 0$

Step 2.2.2.1.4

Factor $3$ out of $3 y^{2} + 3 (4 y)$ .

$3 (y^{2} + 4 y) + 3 \cdot - 5 = 0$

Step 2.2.2.1.5

Factor $3$ out of $3 (y^{2} + 4 y) + 3 \cdot - 5$ .

$3 (y^{2} + 4 y - 5) = 0$

Step 2.2.2.2

Factor.

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Step 2.2.2.2.1

Factor $y^{2} + 4 y - 5$ using the AC method.

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Step 2.2.2.2.1.1

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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 2.2.2.2.1.2

Write the factored form using these integers.

$3 ((y - 1) (y + 5)) = 0$

Step 2.2.2.2.2

Remove unnecessary parentheses.

$3 (y - 1) (y + 5) = 0$

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 $0$ .

$y - 1 = 0$

$y + 5 = 0$

Step 2.2.4

Set $y - 1$ equal to $0$ and solve for $y$ .

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Step 2.2.4.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Step 2.2.4.2

Add $1$ to both sides of the equation.

$y = 1$

Step 2.2.5

Set $y + 5$ equal to $0$ and solve for $y$ .

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Step 2.2.5.1

Set $y + 5$ equal to $0$ .

$y + 5 = 0$

Step 2.2.5.2

Subtract $5$ from both sides of the equation.

$y = - 5$

Step 2.2.6

The final solution is all the values that make $3 (y - 1) (y + 5) = 0$ true.

$y = 1, - 5$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 1), (0, - 5)$

Step 3

List the intersections.

x-intercept(s): $(4 + \sqrt{21}, 0), (4 - \sqrt{21}, 0)$

y-intercept(s): $(0, 1), (0, - 5)$

Step 4