Precalculus Examples

$3 x^{2} - 12 x + 3 y^{2} - 30 y = 12$

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 $x$ .

$3 x^{2} - 12 x + 3 {(0)}^{2} - 30 \cdot 0 = 12$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $3 x^{2} - 12 x + 3 {(0)}^{2} - 30 \cdot 0$ .

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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 $0$ .

$3 x^{2} - 12 x + 3 \cdot 0 - 30 \cdot 0 = 12$

Step 1.2.1.1.2

Multiply $3$ by $0$ .

$3 x^{2} - 12 x + 0 - 30 \cdot 0 = 12$

Step 1.2.1.1.3

Multiply $- 30$ by $0$ .

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

Step 1.2.1.2

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

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Step 1.2.1.2.1

Add $3 x^{2} - 12 x$ and $0$ .

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

Step 1.2.1.2.2

Add $3 x^{2} - 12 x$ and $0$ .

$3 x^{2} - 12 x = 12$

Step 1.2.2

Subtract $12$ from both sides of the equation.

$3 x^{2} - 12 x - 12 = 0$

Step 1.2.3

Factor $3$ out of $3 x^{2} - 12 x - 12$ .

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Step 1.2.3.1

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

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

Step 1.2.3.2

Factor $3$ out of $- 12 x$ .

$3 (x^{2}) + 3 (- 4 x) - 12 = 0$

Step 1.2.3.3

Factor $3$ out of $- 12$ .

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

Step 1.2.3.4

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

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

Step 1.2.3.5

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

$3 (x^{2} - 4 x - 4) = 0$

Step 1.2.4

Divide each term in $3 (x^{2} - 4 x - 4) = 0$ by $3$ and simplify.

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Step 1.2.4.1

Divide each term in $3 (x^{2} - 4 x - 4) = 0$ by $3$ .

$\frac{3 (x^{2} - 4 x - 4)}{3} = \frac{0}{3}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{3 (x^{2} - 4 x - 4)}{3} = \frac{0}{3}$

Step 1.2.4.2.1.2

Divide $x^{2} - 4 x - 4$ by $1$ .

$x^{2} - 4 x - 4 = \frac{0}{3}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Divide $0$ by $3$ .

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

Step 1.2.5

Use the quadratic formula to find the solutions.

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

Step 1.2.6

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 1.2.7

Simplify.

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 1.2.7.1.2

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

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 1.2.7.1.3

Add $16$ and $16$ .

$x = \frac{4 \pm \sqrt{32}}{2 \cdot 1}$

Step 1.2.7.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.2.7.1.4.1

Factor $16$ out of $32$ .

$x = \frac{4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 1.2.7.1.4.2

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

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

Step 1.2.7.1.5

Pull terms out from under the radical.

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

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 4 \sqrt{2}}{2}$

Step 1.2.7.3

Simplify $\frac{4 \pm 4 \sqrt{2}}{2}$ .

$x = 2 \pm 2 \sqrt{2}$

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 $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 1.2.8.1.2

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

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Step 1.2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 1.2.8.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 1.2.8.1.3

Add $16$ and $16$ .

$x = \frac{4 \pm \sqrt{32}}{2 \cdot 1}$

Step 1.2.8.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.2.8.1.4.1

Factor $16$ out of $32$ .

$x = \frac{4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 1.2.8.1.4.2

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

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

Step 1.2.8.1.5

Pull terms out from under the radical.

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

Step 1.2.8.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 4 \sqrt{2}}{2}$

Step 1.2.8.3

Simplify $\frac{4 \pm 4 \sqrt{2}}{2}$ .

$x = 2 \pm 2 \sqrt{2}$

Step 1.2.8.4

Change the $\pm$ to $+$ .

$x = 2 + 2 \sqrt{2}$

Step 1.2.9

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

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Step 1.2.9.1

Simplify the numerator.

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Step 1.2.9.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 1.2.9.1.2

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

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Step 1.2.9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 1.2.9.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 1.2.9.1.3

Add $16$ and $16$ .

$x = \frac{4 \pm \sqrt{32}}{2 \cdot 1}$

Step 1.2.9.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.2.9.1.4.1

Factor $16$ out of $32$ .

$x = \frac{4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 1.2.9.1.4.2

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

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

Step 1.2.9.1.5

Pull terms out from under the radical.

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

Step 1.2.9.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 4 \sqrt{2}}{2}$

Step 1.2.9.3

Simplify $\frac{4 \pm 4 \sqrt{2}}{2}$ .

$x = 2 \pm 2 \sqrt{2}$

Step 1.2.9.4

Change the $\pm$ to $-$ .

$x = 2 - 2 \sqrt{2}$

Step 1.2.10

The final answer is the combination of both solutions.

$x = 2 + 2 \sqrt{2}, 2 - 2 \sqrt{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2 + 2 \sqrt{2}, 0), (2 - 2 \sqrt{2}, 0)$

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 $y$ .

$3 {(0)}^{2} - 12 \cdot 0 + 3 y^{2} - 30 y = 12$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $3 {(0)}^{2} - 12 \cdot 0 + 3 y^{2} - 30 y$ .

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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 $0$ .

$3 \cdot 0 - 12 \cdot 0 + 3 y^{2} - 30 y = 12$

Step 2.2.1.1.2

Multiply $3$ by $0$ .

$0 - 12 \cdot 0 + 3 y^{2} - 30 y = 12$

Step 2.2.1.1.3

Multiply $- 12$ by $0$ .

$0 + 0 + 3 y^{2} - 30 y = 12$

Step 2.2.1.2

Combine the opposite terms in $0 + 0 + 3 y^{2} - 30 y$ .

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Step 2.2.1.2.1

Add $0$ and $0$ .

$0 + 3 y^{2} - 30 y = 12$

Step 2.2.1.2.2

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

$3 y^{2} - 30 y = 12$

Step 2.2.2

Subtract $12$ from both sides of the equation.

$3 y^{2} - 30 y - 12 = 0$

Step 2.2.3

Factor $3$ out of $3 y^{2} - 30 y - 12$ .

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Step 2.2.3.1

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

$3 (y^{2}) - 30 y - 12 = 0$

Step 2.2.3.2

Factor $3$ out of $- 30 y$ .

$3 (y^{2}) + 3 (- 10 y) - 12 = 0$

Step 2.2.3.3

Factor $3$ out of $- 12$ .

$3 y^{2} + 3 (- 10 y) + 3 \cdot - 4 = 0$

Step 2.2.3.4

Factor $3$ out of $3 y^{2} + 3 (- 10 y)$ .

$3 (y^{2} - 10 y) + 3 \cdot - 4 = 0$

Step 2.2.3.5

Factor $3$ out of $3 (y^{2} - 10 y) + 3 \cdot - 4$ .

$3 (y^{2} - 10 y - 4) = 0$

Step 2.2.4

Divide each term in $3 (y^{2} - 10 y - 4) = 0$ by $3$ and simplify.

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Step 2.2.4.1

Divide each term in $3 (y^{2} - 10 y - 4) = 0$ by $3$ .

$\frac{3 (y^{2} - 10 y - 4)}{3} = \frac{0}{3}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $3$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

$\frac{3 (y^{2} - 10 y - 4)}{3} = \frac{0}{3}$

Step 2.2.4.2.1.2

Divide $y^{2} - 10 y - 4$ by $1$ .

$y^{2} - 10 y - 4 = \frac{0}{3}$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Divide $0$ by $3$ .

$y^{2} - 10 y - 4 = 0$

Step 2.2.5

Use the quadratic formula to find the solutions.

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

Step 2.2.6

Substitute the values $a = 1$ , $b = - 10$ , and $c = - 4$ into the quadratic formula and solve for $y$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1}$

Step 2.2.7

Simplify.

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Step 2.2.7.1

Simplify the numerator.

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Step 2.2.7.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 2.2.7.1.2

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

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Step 2.2.7.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot - 4}}{2 \cdot 1}$

Step 2.2.7.1.2.2

Multiply $- 4$ by $- 4$ .

$y = \frac{10 \pm \sqrt{100 + 16}}{2 \cdot 1}$

Step 2.2.7.1.3

Add $100$ and $16$ .

$y = \frac{10 \pm \sqrt{116}}{2 \cdot 1}$

Step 2.2.7.1.4

Rewrite $116$ as $2^{2} \cdot 29$ .

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Step 2.2.7.1.4.1

Factor $4$ out of $116$ .

$y = \frac{10 \pm \sqrt{4 (29)}}{2 \cdot 1}$

Step 2.2.7.1.4.2

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

$y = \frac{10 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Step 2.2.7.1.5

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{29}}{2 \cdot 1}$

Step 2.2.7.2

Multiply $2$ by $1$ .

$y = \frac{10 \pm 2 \sqrt{29}}{2}$

Step 2.2.7.3

Simplify $\frac{10 \pm 2 \sqrt{29}}{2}$ .

$y = 5 \pm \sqrt{29}$

Step 2.2.8

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

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Step 2.2.8.1

Simplify the numerator.

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Step 2.2.8.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 2.2.8.1.2

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

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Step 2.2.8.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot - 4}}{2 \cdot 1}$

Step 2.2.8.1.2.2

Multiply $- 4$ by $- 4$ .

$y = \frac{10 \pm \sqrt{100 + 16}}{2 \cdot 1}$

Step 2.2.8.1.3

Add $100$ and $16$ .

$y = \frac{10 \pm \sqrt{116}}{2 \cdot 1}$

Step 2.2.8.1.4

Rewrite $116$ as $2^{2} \cdot 29$ .

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Step 2.2.8.1.4.1

Factor $4$ out of $116$ .

$y = \frac{10 \pm \sqrt{4 (29)}}{2 \cdot 1}$

Step 2.2.8.1.4.2

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

$y = \frac{10 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Step 2.2.8.1.5

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{29}}{2 \cdot 1}$

Step 2.2.8.2

Multiply $2$ by $1$ .

$y = \frac{10 \pm 2 \sqrt{29}}{2}$

Step 2.2.8.3

Simplify $\frac{10 \pm 2 \sqrt{29}}{2}$ .

$y = 5 \pm \sqrt{29}$

Step 2.2.8.4

Change the $\pm$ to $+$ .

$y = 5 + \sqrt{29}$

Step 2.2.9

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

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Step 2.2.9.1

Simplify the numerator.

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Step 2.2.9.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 2.2.9.1.2

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

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Step 2.2.9.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot - 4}}{2 \cdot 1}$

Step 2.2.9.1.2.2

Multiply $- 4$ by $- 4$ .

$y = \frac{10 \pm \sqrt{100 + 16}}{2 \cdot 1}$

Step 2.2.9.1.3

Add $100$ and $16$ .

$y = \frac{10 \pm \sqrt{116}}{2 \cdot 1}$

Step 2.2.9.1.4

Rewrite $116$ as $2^{2} \cdot 29$ .

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Step 2.2.9.1.4.1

Factor $4$ out of $116$ .

$y = \frac{10 \pm \sqrt{4 (29)}}{2 \cdot 1}$

Step 2.2.9.1.4.2

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

$y = \frac{10 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Step 2.2.9.1.5

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{29}}{2 \cdot 1}$

Step 2.2.9.2

Multiply $2$ by $1$ .

$y = \frac{10 \pm 2 \sqrt{29}}{2}$

Step 2.2.9.3

Simplify $\frac{10 \pm 2 \sqrt{29}}{2}$ .

$y = 5 \pm \sqrt{29}$

Step 2.2.9.4

Change the $\pm$ to $-$ .

$y = 5 - \sqrt{29}$

Step 2.2.10

The final answer is the combination of both solutions.

$y = 5 + \sqrt{29}, 5 - \sqrt{29}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 5 + \sqrt{29}), (0, 5 - \sqrt{29})$

Step 3

List the intersections.

x-intercept(s): $(2 + 2 \sqrt{2}, 0), (2 - 2 \sqrt{2}, 0)$

y-intercept(s): $(0, 5 + \sqrt{29}), (0, 5 - \sqrt{29})$

Step 4