Algebra Examples

$x^{3} - 8 x^{2} + 15 x = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x$ out of $x^{3} - 8 x^{2} + 15 x$ .

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Step 1.1.1

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

$x \cdot x^{2} - 8 x^{2} + 15 x = 0$

Step 1.1.2

Factor $x$ out of $- 8 x^{2}$ .

$x \cdot x^{2} + x (- 8 x) + 15 x = 0$

Step 1.1.3

Factor $x$ out of $15 x$ .

$x \cdot x^{2} + x (- 8 x) + x \cdot 15 = 0$

Step 1.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 8 x)$ .

$x (x^{2} - 8 x) + x \cdot 15 = 0$

Step 1.1.5

Factor $x$ out of $x (x^{2} - 8 x) + x \cdot 15$ .

$x (x^{2} - 8 x + 15) = 0$

Step 1.2

Factor.

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Step 1.2.1

Factor $x^{2} - 8 x + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 1.2.1.2

Write the factored form using these integers.

$x ((x - 5) (x - 3)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$x (x - 5) (x - 3) = 0$

Step 2

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

$x = 0$

$x - 5 = 0$

$x - 3 = 0$

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

Add $5$ to both sides of the equation.

$x = 5$

Step 5

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

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Step 5.1

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

$x - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6

The final solution is all the values that make $x (x - 5) (x - 3) = 0$ true.

$x = 0, 5, 3$

Step 7