Calculus Examples

$p (x) = x^{4} - 6 x^{2} + 5$

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 $x$ .

$0 = x^{4} - 6 x^{2} + 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{4} - 6 x^{2} + 5 = 0$ .

$x^{4} - 6 x^{2} + 5 = 0$

Step 1.2.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 6 u + 5 = 0$

$u = x^{2}$

Step 1.2.3

Factor $u^{2} - 6 u + 5$ using the AC method.

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Step 1.2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $5$ and whose sum is $- 6$ .

$- 5, - 1$

Step 1.2.3.2

Write the factored form using these integers.

$(u - 5) (u - 1) = 0$

Step 1.2.4

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

$u - 5 = 0$

$u - 1 = 0$

Step 1.2.5

Set $u - 5$ equal to $0$ and solve for $u$ .

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Step 1.2.5.1

Set $u - 5$ equal to $0$ .

$u - 5 = 0$

Step 1.2.5.2

Add $5$ to both sides of the equation.

$u = 5$

Step 1.2.6

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

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Step 1.2.6.1

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

$u - 1 = 0$

Step 1.2.6.2

Add $1$ to both sides of the equation.

$u = 1$

Step 1.2.7

The final solution is all the values that make $(u - 5) (u - 1) = 0$ true.

$u = 5, 1$

Step 1.2.8

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 5$

${(x^{2})}^{1} = 1$

Step 1.2.9

Solve the first equation for $x$ .

$x^{2} = 5$

Step 1.2.10

Solve the equation for $x$ .

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Step 1.2.10.1

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

$x = \pm \sqrt{5}$

Step 1.2.10.2

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

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Step 1.2.10.2.1

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

$x = \sqrt{5}$

Step 1.2.10.2.2

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

$x = - \sqrt{5}$

Step 1.2.10.2.3

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

$x = \sqrt{5}, - \sqrt{5}$

Step 1.2.11

Solve the second equation for $x$ .

${(x^{2})}^{1} = 1$

Step 1.2.12

Solve the equation for $x$ .

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Step 1.2.12.1

Remove parentheses.

$x^{2} = 1$

Step 1.2.12.2

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

$x = \pm \sqrt{1}$

Step 1.2.12.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 1.2.12.4

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

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Step 1.2.12.4.1

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

$x = 1$

Step 1.2.12.4.2

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

$x = - 1$

Step 1.2.12.4.3

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

$x = 1, - 1$

Step 1.2.13

The solution to $x^{4} - 6 x^{2} + 5 = 0$ is $x = \sqrt{5}, - \sqrt{5}, 1, - 1$ .

$x = \sqrt{5}, - \sqrt{5}, 1, - 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{5}, 0), (- \sqrt{5}, 0), (1, 0), (- 1, 0)$

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 $y$ .

$y = {(0)}^{4} - 6 {(0)}^{2} + 5$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{4} - 6 {(0)}^{2} + 5$

Step 2.2.2

Remove parentheses.

$y = 0^{4} - 6 \cdot 0^{2} + 5$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{4} - 6 {(0)}^{2} + 5$

Step 2.2.4

Simplify ${(0)}^{4} - 6 {(0)}^{2} + 5$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

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

$y = 0 - 6 {(0)}^{2} + 5$

Step 2.2.4.1.2

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

$y = 0 - 6 \cdot 0 + 5$

Step 2.2.4.1.3

Multiply $- 6$ by $0$ .

$y = 0 + 0 + 5$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 5$

Step 2.2.4.2.2

Add $0$ and $5$ .

$y = 5$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 5)$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{5}, 0), (- \sqrt{5}, 0), (1, 0), (- 1, 0)$

y-intercept(s): $(0, 5)$

Step 4