Calculus Examples

$15 x^{4} - 75 x^{2} + 60 = 0$

Step 1

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

$15 u^{2} - 75 u + 60 = 0$

$u = x^{2}$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $15$ out of $15 u^{2} - 75 u + 60$ .

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Step 2.1.1

Factor $15$ out of $15 u^{2}$ .

$15 (u^{2}) - 75 u + 60 = 0$

Step 2.1.2

Factor $15$ out of $- 75 u$ .

$15 (u^{2}) + 15 (- 5 u) + 60 = 0$

Step 2.1.3

Factor $15$ out of $60$ .

$15 u^{2} + 15 (- 5 u) + 15 \cdot 4 = 0$

Step 2.1.4

Factor $15$ out of $15 u^{2} + 15 (- 5 u)$ .

$15 (u^{2} - 5 u) + 15 \cdot 4 = 0$

Step 2.1.5

Factor $15$ out of $15 (u^{2} - 5 u) + 15 \cdot 4$ .

$15 (u^{2} - 5 u + 4) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $u^{2} - 5 u + 4$ using the AC method.

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Step 2.2.1.1

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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Step 2.2.1.2

Write the factored form using these integers.

$15 ((u - 4) (u - 1)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$15 (u - 4) (u - 1) = 0$

Step 3

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

$u - 4 = 0$

$u - 1 = 0$

Step 4

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 4.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 4.2

Add $4$ to both sides of the equation.

$u = 4$

Step 5

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

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Step 5.1

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

$u - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$u = 1$

Step 6

The final solution is all the values that make $15 (u - 4) (u - 1) = 0$ true.

$u = 4, 1$

Step 7

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

$x^{2} = 4$

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

Step 8

Solve the first equation for $x$ .

$x^{2} = 4$

Step 9

Solve the equation for $x$ .

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Step 9.1

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

$x = \pm \sqrt{4}$

Step 9.2

Simplify $\pm \sqrt{4}$ .

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Step 9.2.1

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

$x = \pm \sqrt{2^{2}}$

Step 9.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 9.3

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

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Step 9.3.1

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

$x = 2$

Step 9.3.2

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

$x = - 2$

Step 9.3.3

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

$x = 2, - 2$

Step 10

Solve the second equation for $x$ .

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

Step 11

Solve the equation for $x$ .

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Step 11.1

Remove parentheses.

$x^{2} = 1$

Step 11.2

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

$x = \pm \sqrt{1}$

Step 11.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 11.4

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

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Step 11.4.1

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

$x = 1$

Step 11.4.2

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

$x = - 1$

Step 11.4.3

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

$x = 1, - 1$

Step 12

The solution to $15 x^{4} - 75 x^{2} + 60 = 0$ is $x = 2, - 2, 1, - 1$ .

$x = 2, - 2, 1, - 1$