Algebra Examples

$f (x) = x^{3} - 13 x^{2} + 40 x$

Step 1

Set $x^{3} - 13 x^{2} + 40 x$ equal to $0$ .

$x^{3} - 13 x^{2} + 40 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $x$ out of $x^{3} - 13 x^{2} + 40 x$ .

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Step 2.1.1.1

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

$x \cdot x^{2} - 13 x^{2} + 40 x = 0$

Step 2.1.1.2

Factor $x$ out of $- 13 x^{2}$ .

$x \cdot x^{2} + x (- 13 x) + 40 x = 0$

Step 2.1.1.3

Factor $x$ out of $40 x$ .

$x \cdot x^{2} + x (- 13 x) + x \cdot 40 = 0$

Step 2.1.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 13 x)$ .

$x (x^{2} - 13 x) + x \cdot 40 = 0$

Step 2.1.1.5

Factor $x$ out of $x (x^{2} - 13 x) + x \cdot 40$ .

$x (x^{2} - 13 x + 40) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor $x^{2} - 13 x + 40$ using the AC method.

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Step 2.1.2.1.1

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 $40$ and whose sum is $- 13$ .

$- 8, - 5$

Step 2.1.2.1.2

Write the factored form using these integers.

$x ((x - 8) (x - 5)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$x (x - 8) (x - 5) = 0$

Step 2.2

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

$x = 0$

$x - 8 = 0$

$x - 5 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 2.4.2

Add $8$ to both sides of the equation.

$x = 8$

Step 2.5

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

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Step 2.5.1

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

$x - 5 = 0$

Step 2.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.6

The final solution is all the values that make $x (x - 8) (x - 5) = 0$ true.

$x = 0, 8, 5$

Step 3