Calculus Examples

$- 60 x^{4} + 300 x^{3} - 360 x^{2} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $- 60 x^{2}$ out of $- 60 x^{4} + 300 x^{3} - 360 x^{2}$ .

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Step 1.1.1

Factor $- 60 x^{2}$ out of $- 60 x^{4}$ .

$- 60 x^{2} x^{2} + 300 x^{3} - 360 x^{2} = 0$

Step 1.1.2

Factor $- 60 x^{2}$ out of $300 x^{3}$ .

$- 60 x^{2} x^{2} - 60 x^{2} (- 5 x) - 360 x^{2} = 0$

Step 1.1.3

Factor $- 60 x^{2}$ out of $- 360 x^{2}$ .

$- 60 x^{2} x^{2} - 60 x^{2} (- 5 x) - 60 x^{2} \cdot 6 = 0$

Step 1.1.4

Factor $- 60 x^{2}$ out of $- 60 x^{2} (x^{2}) - 60 x^{2} (- 5 x)$ .

$- 60 x^{2} (x^{2} - 5 x) - 60 x^{2} \cdot 6 = 0$

Step 1.1.5

Factor $- 60 x^{2}$ out of $- 60 x^{2} (x^{2} - 5 x) - 60 x^{2} (6)$ .

$- 60 x^{2} (x^{2} - 5 x + 6) = 0$

Step 1.2

Factor.

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Step 1.2.1

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

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Step 1.2.1.1

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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 1.2.1.2

Write the factored form using these integers.

$- 60 x^{2} ((x - 3) (x - 2)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$- 60 x^{2} (x - 3) (x - 2) = 0$

Step 2

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

$x^{2} = 0$

$x - 3 = 0$

$x - 2 = 0$

Step 3

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 3.2

Solve $x^{2} = 0$ for $x$ .

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Step 3.2.1

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

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

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

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Step 5.1

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

$x - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6

The final solution is all the values that make $- 60 x^{2} (x - 3) (x - 2) = 0$ true.

$x = 0, 3, 2$