Calculus Examples

$y = (x - 2) \sqrt{x^{2} + 1}$

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(x - 2) \sqrt{x^{2} + 1} = 0$ .

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

Step 1.2.2

Apply the distributive property.

$x \sqrt{x^{2} + 1} - 2 \sqrt{x^{2} + 1} = 0$

Step 1.2.3

Simplify the left side.

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Step 1.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 1.2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 1.2.4

Factor the left side of the equation.

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Step 1.2.4.1

Let $u = {(x^{2} + 1)}^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of ${(x^{2} + 1)}^{\frac{1}{2}}$ .

$x u - 2 u = 0$

Step 1.2.4.2

Factor $u$ out of $x u - 2 u$ .

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Step 1.2.4.2.1

Factor $u$ out of $x u$ .

$u x - 2 u = 0$

Step 1.2.4.2.2

Factor $u$ out of $- 2 u$ .

$u x + u \cdot - 2 = 0$

Step 1.2.4.2.3

Factor $u$ out of $u x + u \cdot - 2$ .

$u (x - 2) = 0$

Step 1.2.4.3

Replace all occurrences of $u$ with ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 1.2.5

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

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

$x - 2 = 0$

Step 1.2.6

Set ${(x^{2} + 1)}^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set ${(x^{2} + 1)}^{\frac{1}{2}}$ equal to $0$ .

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

Step 1.2.6.2

Solve ${(x^{2} + 1)}^{\frac{1}{2}} = 0$ for $x$ .

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Step 1.2.6.2.1

Set the $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 1.2.6.2.2

Solve for $x$ .

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Step 1.2.6.2.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 1.2.6.2.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 1.2.6.2.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 1.2.6.2.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.6.2.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 1.2.6.2.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 1.2.6.2.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 1.2.7

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.8

The final solution is all the values that make ${(x^{2} + 1)}^{\frac{1}{2}} (x - 2) = 0$ true.

$x = i, - i, 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Simplify $((0) - 2) \sqrt{{(0)}^{2} + 1}$ .

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Step 2.2.4.1

Subtract $2$ from $0$ .

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

Step 2.2.4.2

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

$y = - 2 \sqrt{0 + 1}$

Step 2.2.4.3

Add $0$ and $1$ .

$y = - 2 \sqrt{1}$

Step 2.2.4.4

Any root of $1$ is $1$ .

$y = - 2 \cdot 1$

Step 2.2.4.5

Multiply $- 2$ by $1$ .

$y = - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(2, 0)$

y-intercept(s): $(0, - 2)$

Step 4