Calculus Examples

$y = x^{4} - 8 x^{2} + 8$

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 $x$ .

$0 = x^{4} - 8 x^{2} + 8$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{4} - 8 x^{2} + 8 = 0$ .

$x^{4} - 8 x^{2} + 8 = 0$

Step 1.2.2

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

$u^{2} - 8 u + 8 = 0$

$u = x^{2}$

Step 1.2.3

Use the quadratic formula to find the solutions.

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

Step 1.2.4

Substitute the values $a = 1$ , $b = - 8$ , and $c = 8$ into the quadratic formula and solve for $u$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 8)}}{2 \cdot 1}$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

Raise $- 8$ to the power of $2$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 8}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply $- 4$ by $8$ .

$u = \frac{8 \pm \sqrt{64 - 32}}{2 \cdot 1}$

Step 1.2.5.1.3

Subtract $32$ from $64$ .

$u = \frac{8 \pm \sqrt{32}}{2 \cdot 1}$

Step 1.2.5.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.2.5.1.4.1

Factor $16$ out of $32$ .

$u = \frac{8 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 1.2.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$u = \frac{8 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 1.2.5.1.5

Pull terms out from under the radical.

$u = \frac{8 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 1.2.5.2

Multiply $2$ by $1$ .

$u = \frac{8 \pm 4 \sqrt{2}}{2}$

Step 1.2.5.3

Simplify $\frac{8 \pm 4 \sqrt{2}}{2}$ .

$u = 4 \pm 2 \sqrt{2}$

Step 1.2.6

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

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $- 8$ to the power of $2$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 8}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $8$ .

$u = \frac{8 \pm \sqrt{64 - 32}}{2 \cdot 1}$

Step 1.2.6.1.3

Subtract $32$ from $64$ .

$u = \frac{8 \pm \sqrt{32}}{2 \cdot 1}$

Step 1.2.6.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.2.6.1.4.1

Factor $16$ out of $32$ .

$u = \frac{8 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 1.2.6.1.4.2

Rewrite $16$ as $4^{2}$ .

$u = \frac{8 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$u = \frac{8 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$u = \frac{8 \pm 4 \sqrt{2}}{2}$

Step 1.2.6.3

Simplify $\frac{8 \pm 4 \sqrt{2}}{2}$ .

$u = 4 \pm 2 \sqrt{2}$

Step 1.2.6.4

Change the $\pm$ to $+$ .

$u = 4 + 2 \sqrt{2}$

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 $- 8$ to the power of $2$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 8}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $8$ .

$u = \frac{8 \pm \sqrt{64 - 32}}{2 \cdot 1}$

Step 1.2.7.1.3

Subtract $32$ from $64$ .

$u = \frac{8 \pm \sqrt{32}}{2 \cdot 1}$

Step 1.2.7.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.2.7.1.4.1

Factor $16$ out of $32$ .

$u = \frac{8 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 1.2.7.1.4.2

Rewrite $16$ as $4^{2}$ .

$u = \frac{8 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$u = \frac{8 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$u = \frac{8 \pm 4 \sqrt{2}}{2}$

Step 1.2.7.3

Simplify $\frac{8 \pm 4 \sqrt{2}}{2}$ .

$u = 4 \pm 2 \sqrt{2}$

Step 1.2.7.4

Change the $\pm$ to $-$ .

$u = 4 - 2 \sqrt{2}$

Step 1.2.8

The final answer is the combination of both solutions.

$u = 4 + 2 \sqrt{2}, 4 - 2 \sqrt{2}$

Step 1.2.9

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

$x^{2} = 6.82842712$

${(x^{2})}^{1} = 1.17157287$

Step 1.2.10

Solve the first equation for $x$ .

$x^{2} = 6.82842712$

Step 1.2.11

Solve the equation for $x$ .

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Step 1.2.11.1

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

$x = \pm \sqrt{6.82842712}$

Step 1.2.11.2

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

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Step 1.2.11.2.1

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

$x = \sqrt{6.82842712}$

Step 1.2.11.2.2

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

$x = - \sqrt{6.82842712}$

Step 1.2.11.2.3

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

$x = \sqrt{6.82842712}, - \sqrt{6.82842712}$

Step 1.2.12

Solve the second equation for $x$ .

${(x^{2})}^{1} = 1.17157287$

Step 1.2.13

Solve the equation for $x$ .

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Step 1.2.13.1

Remove parentheses.

$x^{2} = 1.17157287$

Step 1.2.13.2

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

$x = \pm \sqrt{1.17157287}$

Step 1.2.13.3

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

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Step 1.2.13.3.1

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

$x = \sqrt{1.17157287}$

Step 1.2.13.3.2

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

$x = - \sqrt{1.17157287}$

Step 1.2.13.3.3

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

$x = \sqrt{1.17157287}, - \sqrt{1.17157287}$

Step 1.2.14

The solution to $x^{4} - 8 x^{2} + 8 = 0$ is $x = \sqrt{6.82842712}, - \sqrt{6.82842712}, \sqrt{1.17157287}, - \sqrt{1.17157287}$ .

$x = \sqrt{6.82842712}, - \sqrt{6.82842712}, \sqrt{1.17157287}, - \sqrt{1.17157287}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{6.82842712}, 0), (- \sqrt{6.82842712}, 0), (\sqrt{1.17157287}, 0), (- \sqrt{1.17157287}, 0)$

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 $y$ .

$y = {(0)}^{4} - 8 {(0)}^{2} + 8$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{4} - 8 {(0)}^{2} + 8$

Step 2.2.2

Remove parentheses.

$y = 0^{4} - 8 \cdot 0^{2} + 8$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{4} - 8 {(0)}^{2} + 8$

Step 2.2.4

Simplify ${(0)}^{4} - 8 {(0)}^{2} + 8$ .

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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 $0$ .

$y = 0 - 8 {(0)}^{2} + 8$

Step 2.2.4.1.2

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

$y = 0 - 8 \cdot 0 + 8$

Step 2.2.4.1.3

Multiply $- 8$ by $0$ .

$y = 0 + 0 + 8$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 8$

Step 2.2.4.2.2

Add $0$ and $8$ .

$y = 8$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 8)$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{6.82842712}, 0), (- \sqrt{6.82842712}, 0), (\sqrt{1.17157287}, 0), (- \sqrt{1.17157287}, 0)$

y-intercept(s): $(0, 8)$

Step 4