Finite Math Examples

$f (x) = - 3 x^{2} + 6 x + 6$

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 $x$ .

$0 = - 3 x^{2} + 6 x + 6$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 3 x^{2} + 6 x + 6 = 0$ .

$- 3 x^{2} + 6 x + 6 = 0$

Step 1.2.2

Factor $- 3$ out of $- 3 x^{2} + 6 x + 6$ .

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Step 1.2.2.1

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

$- 3 x^{2} + 6 x + 6 = 0$

Step 1.2.2.2

Factor $- 3$ out of $6 x$ .

$- 3 x^{2} - 3 (- 2 x) + 6 = 0$

Step 1.2.2.3

Factor $- 3$ out of $6$ .

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

Step 1.2.2.4

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

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

Step 1.2.2.5

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

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

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.

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

Step 1.2.3.2.1.2

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

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

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $- 3$ .

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

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 $a = 1$ , $b = - 2$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

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

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

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

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.6.1.2.2

Multiply $- 4$ by $- 2$ .

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

Step 1.2.6.1.3

Add $4$ and $8$ .

$x = \frac{2 \pm \sqrt{12}}{2 \cdot 1}$

Step 1.2.6.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.6.1.4.1

Factor $4$ out of $12$ .

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

Step 1.2.6.1.4.2

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

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

Step 1.2.6.1.5

Pull terms out from under the radical.

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

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{3}}{2}$

Step 1.2.6.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{2}$ .

$x = 1 \pm \sqrt{3}$

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

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

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.7.1.2.2

Multiply $- 4$ by $- 2$ .

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

Step 1.2.7.1.3

Add $4$ and $8$ .

$x = \frac{2 \pm \sqrt{12}}{2 \cdot 1}$

Step 1.2.7.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.7.1.4.1

Factor $4$ out of $12$ .

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

Step 1.2.7.1.4.2

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

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

Step 1.2.7.1.5

Pull terms out from under the radical.

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

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{3}}{2}$

Step 1.2.7.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{2}$ .

$x = 1 \pm \sqrt{3}$

Step 1.2.7.4

Change the $\pm$ to $+$ .

$x = 1 + \sqrt{3}$

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

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

Step 1.2.8.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.8.1.2.2

Multiply $- 4$ by $- 2$ .

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

Step 1.2.8.1.3

Add $4$ and $8$ .

$x = \frac{2 \pm \sqrt{12}}{2 \cdot 1}$

Step 1.2.8.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.8.1.4.1

Factor $4$ out of $12$ .

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

Step 1.2.8.1.4.2

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

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

Step 1.2.8.1.5

Pull terms out from under the radical.

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

Step 1.2.8.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{3}}{2}$

Step 1.2.8.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{2}$ .

$x = 1 \pm \sqrt{3}$

Step 1.2.8.4

Change the $\pm$ to $-$ .

$x = 1 - \sqrt{3}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = 1 + \sqrt{3}, 1 - \sqrt{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1 + \sqrt{3}, 0), (1 - \sqrt{3}, 0)$

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 $y$ .

$y = - 3 {(0)}^{2} + 6 (0) + 6$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 3 \cdot 0^{2} + 6 (0) + 6$

Step 2.2.2

Remove parentheses.

$y = - 3 {(0)}^{2} + 6 (0) + 6$

Step 2.2.3

Simplify $- 3 {(0)}^{2} + 6 (0) + 6$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = - 3 \cdot 0 + 6 (0) + 6$

Step 2.2.3.1.2

Multiply $- 3$ by $0$ .

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

Step 2.2.3.1.3

Multiply $6$ by $0$ .

$y = 0 + 0 + 6$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 6$

Step 2.2.3.2.2

Add $0$ and $6$ .

$y = 6$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(1 + \sqrt{3}, 0), (1 - \sqrt{3}, 0)$

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

Step 4