Calculus Examples

$y = 18 x - 9 x^{2} - 9$

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 $x$ .

$0 = 18 x - 9 x^{2} - 9$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $18 x - 9 x^{2} - 9 = 0$ .

$18 x - 9 x^{2} - 9 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 9$ out of $18 x - 9 x^{2} - 9$ .

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Step 1.2.2.1.1

Reorder $18 x$ and $- 9 x^{2}$ .

$- 9 x^{2} + 18 x - 9 = 0$

Step 1.2.2.1.2

Factor $- 9$ out of $- 9 x^{2}$ .

$- 9 x^{2} + 18 x - 9 = 0$

Step 1.2.2.1.3

Factor $- 9$ out of $18 x$ .

$- 9 x^{2} - 9 (- 2 x) - 9 = 0$

Step 1.2.2.1.4

Factor $- 9$ out of $- 9$ .

$- 9 x^{2} - 9 (- 2 x) - 9 \cdot 1 = 0$

Step 1.2.2.1.5

Factor $- 9$ out of $- 9 (x^{2}) - 9 (- 2 x)$ .

$- 9 (x^{2} - 2 x) - 9 \cdot 1 = 0$

Step 1.2.2.1.6

Factor $- 9$ out of $- 9 (x^{2} - 2 x) - 9 (1)$ .

$- 9 (x^{2} - 2 x + 1) = 0$

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

Rewrite $1$ as $1^{2}$ .

$- 9 (x^{2} - 2 x + 1^{2}) = 0$

Step 1.2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.2.2.2.3

Rewrite the polynomial.

$- 9 (x^{2} - 2 \cdot x \cdot 1 + 1^{2}) = 0$

Step 1.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$- 9 {(x - 1)}^{2} = 0$

Step 1.2.3

Divide each term in $- 9 {(x - 1)}^{2} = 0$ by $- 9$ and simplify.

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Step 1.2.3.1

Divide each term in $- 9 {(x - 1)}^{2} = 0$ by $- 9$ .

$\frac{- 9 {(x - 1)}^{2}}{- 9} = \frac{0}{- 9}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 9 {(x - 1)}^{2}}{- 9} = \frac{0}{- 9}$

Step 1.2.3.2.1.2

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

${(x - 1)}^{2} = \frac{0}{- 9}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $- 9$ .

${(x - 1)}^{2} = 0$

Step 1.2.4

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.5

Add $1$ to both sides of the equation.

$x = 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1, 0)$

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 $y$ .

$y = 18 (0) - 9 {(0)}^{2} - 9$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 18 (0) - 9 \cdot 0^{2} - 9$

Step 2.2.2

Remove parentheses.

$y = 18 (0) - 9 {(0)}^{2} - 9$

Step 2.2.3

Simplify $18 (0) - 9 {(0)}^{2} - 9$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $18$ by $0$ .

$y = 0 - 9 {(0)}^{2} - 9$

Step 2.2.3.1.2

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

$y = 0 - 9 \cdot 0 - 9$

Step 2.2.3.1.3

Multiply $- 9$ by $0$ .

$y = 0 + 0 - 9$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 9$

Step 2.2.3.2.2

Subtract $9$ from $0$ .

$y = - 9$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 9)$

Step 3

List the intersections.

x-intercept(s): $(1, 0)$

y-intercept(s): $(0, - 9)$

Step 4