Calculus Examples

$y = x - x^{2}$

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 $x$ .

$0 = x - x^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$x - x^{2} = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$u - u^{2} = 0$

Step 1.2.2.2

Factor $u$ out of $u - u^{2}$ .

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Step 1.2.2.2.1

Raise $u$ to the power of $1$ .

$u - u^{2} = 0$

Step 1.2.2.2.2

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - u^{2} = 0$

Step 1.2.2.2.3

Factor $u$ out of $- u^{2}$ .

$u \cdot 1 + u (- u) = 0$

Step 1.2.2.2.4

Factor $u$ out of $u \cdot 1 + u (- u)$ .

$u (1 - u) = 0$

Step 1.2.2.3

Replace all occurrences of $u$ with $x$ .

$x (1 - x) = 0$

Step 1.2.3

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

$x = 0$

$1 - x = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

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

$1 - x = 0$

Step 1.2.5.2

Solve $1 - x = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 1.2.5.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

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

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.5.2.2.2.2

Divide $x$ by $1$ .

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

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 1.2.6

The final solution is all the values that make $x (1 - x) = 0$ true.

$x = 0, 1$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0 - 0^{2}$

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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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 $0$ .

$y = 0 - 0$

Step 2.2.3.1.2

Multiply $- 1$ by $0$ .

$y = 0 + 0$

Step 2.2.3.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4