Calculus Examples

$- 4 x^{3} + 100 x = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $- 4 x$ out of $- 4 x^{3} + 100 x$ .

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Step 1.1.1

Factor $- 4 x$ out of $- 4 x^{3}$ .

$- 4 x \cdot x^{2} + 100 x = 0$

Step 1.1.2

Factor $- 4 x$ out of $100 x$ .

$- 4 x \cdot x^{2} - 4 x \cdot - 25 = 0$

Step 1.1.3

Factor $- 4 x$ out of $- 4 x (x^{2}) - 4 x (- 25)$ .

$- 4 x (x^{2} - 25) = 0$

Step 1.2

Rewrite $25$ as $5^{2}$ .

$- 4 x (x^{2} - 5^{2}) = 0$

Step 1.3

Factor.

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Step 1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$- 4 x ((x + 5) (x - 5)) = 0$

Step 1.3.2

Remove unnecessary parentheses.

$- 4 x (x + 5) (x - 5) = 0$

Step 2

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

$x = 0$

$x + 5 = 0$

$x - 5 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 4.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 5

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

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Step 5.1

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

$x - 5 = 0$

Step 5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 6

The final solution is all the values that make $- 4 x (x + 5) (x - 5) = 0$ true.

$x = 0, - 5, 5$