Calculus Examples

$y = (x - 6) (x + 5) (6 x - 3)$

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 $x$ .

$0 = (x - 6) (x + 5) (6 x - 3)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(x - 6) (x + 5) (6 x - 3) = 0$ .

$(x - 6) (x + 5) (6 x - 3) = 0$

Step 1.2.2

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

$x - 6 = 0$

$x + 5 = 0$

$6 x - 3 = 0$

Step 1.2.3

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

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Step 1.2.3.1

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

$x - 6 = 0$

Step 1.2.3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 1.2.4

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

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Step 1.2.4.1

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

$x + 5 = 0$

Step 1.2.4.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 1.2.5

Set $6 x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $6 x - 3$ equal to $0$ .

$6 x - 3 = 0$

Step 1.2.5.2

Solve $6 x - 3 = 0$ for $x$ .

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Step 1.2.5.2.1

Add $3$ to both sides of the equation.

$6 x = 3$

Step 1.2.5.2.2

Divide each term in $6 x = 3$ by $6$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $6 x = 3$ by $6$ .

$\frac{6 x}{6} = \frac{3}{6}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{3}{6}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{6}$

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Cancel the common factor of $3$ and $6$ .

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Step 1.2.5.2.2.3.1.1

Factor $3$ out of $3$ .

$x = \frac{3 (1)}{6}$

Step 1.2.5.2.2.3.1.2

Cancel the common factors.

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Step 1.2.5.2.2.3.1.2.1

Factor $3$ out of $6$ .

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

Step 1.2.5.2.2.3.1.2.2

Cancel the common factor.

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

Step 1.2.5.2.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{2}$

Step 1.2.6

The final solution is all the values that make $(x - 6) (x + 5) (6 x - 3) = 0$ true.

$x = 6, - 5, \frac{1}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(6, 0), (- 5, 0), (\frac{1}{2}, 0)$

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 $y$ .

$y = ((0) - 6) ((0) + 5) (6 (0) - 3)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = (0 - 6) ((0) + 5) (6 (0) - 3)$

Step 2.2.2

Remove parentheses.

$y = (0 - 6) (0 + 5) (6 (0) - 3)$

Step 2.2.3

Remove parentheses.

$y = ((0) - 6) ((0) + 5) (6 (0) - 3)$

Step 2.2.4

Simplify $((0) - 6) ((0) + 5) (6 (0) - 3)$ .

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Step 2.2.4.1

Subtract $6$ from $0$ .

$y = - 6 ((0) + 5) (6 (0) - 3)$

Step 2.2.4.2

Add $0$ and $5$ .

$y = - 6 \cdot 5 (6 (0) - 3)$

Step 2.2.4.3

Multiply $- 6$ by $5$ .

$y = - 30 (6 (0) - 3)$

Step 2.2.4.4

Multiply $6$ by $0$ .

$y = - 30 (0 - 3)$

Step 2.2.4.5

Subtract $3$ from $0$ .

$y = - 30 \cdot - 3$

Step 2.2.4.6

Multiply $- 30$ by $- 3$ .

$y = 90$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 90)$

Step 3

List the intersections.

x-intercept(s): $(6, 0), (- 5, 0), (\frac{1}{2}, 0)$

y-intercept(s): $(0, 90)$

Step 4