Calculus Examples

$x + y^{2} - y = 1$

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 $x$ .

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

Step 1.2

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

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Step 1.2.1

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

$x + 0 - (0) = 1$

Step 1.2.2

Add $x$ and $0$ .

$x - (0) = 1$

Step 1.2.3

Subtract $0$ from $x$ .

$x = 1$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Add $0$ and $y^{2}$ .

$y^{2} - y = 1$

Step 2.2.2

Subtract $1$ from both sides of the equation.

$y^{2} - y - 1 = 0$

Step 2.2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2.4

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 1$ into the quadratic formula and solve for $y$ .

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

Step 2.2.5

Simplify.

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.2.5.1.2

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

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Step 2.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.5.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 2.2.5.1.3

Add $1$ and $4$ .

$y = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

Step 2.2.5.2

Multiply $2$ by $1$ .

$y = \frac{1 \pm \sqrt{5}}{2}$

Step 2.2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.2.6.1.2

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

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Step 2.2.6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.6.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 2.2.6.1.3

Add $1$ and $4$ .

$y = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

Step 2.2.6.2

Multiply $2$ by $1$ .

$y = \frac{1 \pm \sqrt{5}}{2}$

Step 2.2.6.3

Change the $\pm$ to $+$ .

$y = \frac{1 + \sqrt{5}}{2}$

Step 2.2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.2.7.1

Simplify the numerator.

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Step 2.2.7.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.2.7.1.2

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

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Step 2.2.7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.7.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 2.2.7.1.3

Add $1$ and $4$ .

$y = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

Step 2.2.7.2

Multiply $2$ by $1$ .

$y = \frac{1 \pm \sqrt{5}}{2}$

Step 2.2.7.3

Change the $\pm$ to $-$ .

$y = \frac{1 - \sqrt{5}}{2}$

Step 2.2.8

The final answer is the combination of both solutions.

$y = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1 + \sqrt{5}}{2}), (0, \frac{1 - \sqrt{5}}{2})$

Step 3

List the intersections.

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

y-intercept(s): $(0, \frac{1 + \sqrt{5}}{2}), (0, \frac{1 - \sqrt{5}}{2})$

Step 4