Calculus Examples

$y = \frac{1}{5} x^{2} - \frac{12}{5} x + \frac{1}{5}$

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 $x$ .

$0 = \frac{1}{5} x^{2} - \frac{12}{5} x + \frac{1}{5}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{5} x^{2} - \frac{12}{5} x + \frac{1}{5} = 0$ .

$\frac{1}{5} x^{2} - \frac{12}{5} x + \frac{1}{5} = 0$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Combine $\frac{1}{5}$ and $x^{2}$ .

$\frac{x^{2}}{5} - \frac{12}{5} x + \frac{1}{5} = 0$

Step 1.2.2.2

Combine $x$ and $\frac{12}{5}$ .

$\frac{x^{2}}{5} - \frac{x \cdot 12}{5} + \frac{1}{5} = 0$

Step 1.2.2.3

Move $12$ to the left of $x$ .

$\frac{x^{2}}{5} - \frac{12 x}{5} + \frac{1}{5} = 0$

Step 1.2.3

Multiply through by the least common denominator $5$ , then simplify.

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Step 1.2.3.1

Apply the distributive property.

$5 (\frac{x^{2}}{5}) + 5 (- \frac{12 x}{5}) + 5 (\frac{1}{5}) = 0$

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Cancel the common factor of $5$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$5 (\frac{x^{2}}{5}) + 5 (- \frac{12 x}{5}) + 5 (\frac{1}{5}) = 0$

Step 1.2.3.2.1.2

Rewrite the expression.

$x^{2} + 5 (- \frac{12 x}{5}) + 5 (\frac{1}{5}) = 0$

Step 1.2.3.2.2

Cancel the common factor of $5$ .

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Step 1.2.3.2.2.1

Move the leading negative in $- \frac{12 x}{5}$ into the numerator.

$x^{2} + 5 (\frac{- 12 x}{5}) + 5 (\frac{1}{5}) = 0$

Step 1.2.3.2.2.2

Cancel the common factor.

$x^{2} + 5 (\frac{- 12 x}{5}) + 5 (\frac{1}{5}) = 0$

Step 1.2.3.2.2.3

Rewrite the expression.

$x^{2} - 12 x + 5 (\frac{1}{5}) = 0$

Step 1.2.3.2.3

Cancel the common factor of $5$ .

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Step 1.2.3.2.3.1

Cancel the common factor.

$x^{2} - 12 x + 5 (\frac{1}{5}) = 0$

Step 1.2.3.2.3.2

Rewrite the expression.

$x^{2} - 12 x + 1 = 0$

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 $a = 1$ , $b = - 12$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

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 $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 1.2.6.1.2

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

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4}}{2 \cdot 1}$

Step 1.2.6.1.3

Subtract $4$ from $144$ .

$x = \frac{12 \pm \sqrt{140}}{2 \cdot 1}$

Step 1.2.6.1.4

Rewrite $140$ as $2^{2} \cdot 35$ .

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Step 1.2.6.1.4.1

Factor $4$ out of $140$ .

$x = \frac{12 \pm \sqrt{4 (35)}}{2 \cdot 1}$

Step 1.2.6.1.4.2

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 1}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{35}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{35}}{2}$

Step 1.2.6.3

Simplify $\frac{12 \pm 2 \sqrt{35}}{2}$ .

$x = 6 \pm \sqrt{35}$

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 $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 1.2.7.1.2

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

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4}}{2 \cdot 1}$

Step 1.2.7.1.3

Subtract $4$ from $144$ .

$x = \frac{12 \pm \sqrt{140}}{2 \cdot 1}$

Step 1.2.7.1.4

Rewrite $140$ as $2^{2} \cdot 35$ .

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Step 1.2.7.1.4.1

Factor $4$ out of $140$ .

$x = \frac{12 \pm \sqrt{4 (35)}}{2 \cdot 1}$

Step 1.2.7.1.4.2

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 1}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{35}}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{35}}{2}$

Step 1.2.7.3

Simplify $\frac{12 \pm 2 \sqrt{35}}{2}$ .

$x = 6 \pm \sqrt{35}$

Step 1.2.7.4

Change the $\pm$ to $+$ .

$x = 6 + \sqrt{35}$

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 $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 1.2.8.1.2

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

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Step 1.2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1}}{2 \cdot 1}$

Step 1.2.8.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4}}{2 \cdot 1}$

Step 1.2.8.1.3

Subtract $4$ from $144$ .

$x = \frac{12 \pm \sqrt{140}}{2 \cdot 1}$

Step 1.2.8.1.4

Rewrite $140$ as $2^{2} \cdot 35$ .

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Step 1.2.8.1.4.1

Factor $4$ out of $140$ .

$x = \frac{12 \pm \sqrt{4 (35)}}{2 \cdot 1}$

Step 1.2.8.1.4.2

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 1}$

Step 1.2.8.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{35}}{2 \cdot 1}$

Step 1.2.8.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{35}}{2}$

Step 1.2.8.3

Simplify $\frac{12 \pm 2 \sqrt{35}}{2}$ .

$x = 6 \pm \sqrt{35}$

Step 1.2.8.4

Change the $\pm$ to $-$ .

$x = 6 - \sqrt{35}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = 6 + \sqrt{35}, 6 - \sqrt{35}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(6 + \sqrt{35}, 0), (6 - \sqrt{35}, 0)$

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 $y$ .

$y = \frac{1}{5} \cdot {(0)}^{2} - \frac{12}{5} \cdot 0 + \frac{1}{5}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{5} \cdot 0^{2} - \frac{12}{5} \cdot 0 + \frac{1}{5}$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{5} \cdot {(0)}^{2} - \frac{12}{5} \cdot 0 + \frac{1}{5}$

Step 2.2.3

Simplify $\frac{1}{5} \cdot {(0)}^{2} - \frac{12}{5} \cdot 0 + \frac{1}{5}$ .

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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 $0$ .

$y = \frac{1}{5} \cdot 0 - \frac{12}{5} \cdot 0 + \frac{1}{5}$

Step 2.2.3.1.2

Multiply $\frac{1}{5}$ by $0$ .

$y = 0 - \frac{12}{5} \cdot 0 + \frac{1}{5}$

Step 2.2.3.1.3

Multiply $- \frac{12}{5} \cdot 0$ .

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Step 2.2.3.1.3.1

Multiply $0$ by $- 1$ .

$y = 0 + 0 (\frac{12}{5}) + \frac{1}{5}$

Step 2.2.3.1.3.2

Multiply $0$ by $\frac{12}{5}$ .

$y = 0 + 0 + \frac{1}{5}$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + \frac{1}{5}$

Step 2.2.3.2.2

Add $0$ and $\frac{1}{5}$ .

$y = \frac{1}{5}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1}{5})$

Step 3

List the intersections.

x-intercept(s): $(6 + \sqrt{35}, 0), (6 - \sqrt{35}, 0)$

y-intercept(s): $(0, \frac{1}{5})$

Step 4