Algebra Examples

$5 n^{3} - 30 n^{2} + 40 n = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor

$5 n$ out of

$5 n^{3} - 30 n^{2} + 40 n$ .

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Step 1.1.1

Factor

$5 n$ out of

$5 n^{3}$ .

$5 n (n^{2}) - 30 n^{2} + 40 n = 0$

Step 1.1.2

Factor

$5 n$ out of

$- 30 n^{2}$ .

$5 n (n^{2}) + 5 n (- 6 n) + 40 n = 0$

Step 1.1.3

Factor

$5 n$ out of

$40 n$ .

$5 n (n^{2}) + 5 n (- 6 n) + 5 n (8) = 0$

Step 1.1.4

Factor

$5 n$ out of

$5 n (n^{2}) + 5 n (- 6 n)$ .

$5 n (n^{2} - 6 n) + 5 n (8) = 0$

Step 1.1.5

Factor

$5 n$ out of

$5 n (n^{2} - 6 n) + 5 n (8)$ .

$5 n (n^{2} - 6 n + 8) = 0$

$5 n (n^{2} - 6 n + 8) = 0$

Step 1.2

Factor.

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Step 1.2.1

Factor

$n^{2} - 6 n + 8$ 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

$8$ and whose sum is

$- 6$ .

$- 4, - 2$

Step 1.2.1.2

Write the factored form using these integers.

$5 n ((n - 4) (n - 2)) = 0$

$5 n ((n - 4) (n - 2)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$5 n (n - 4) (n - 2) = 0$

$5 n (n - 4) (n - 2) = 0$

$5 n (n - 4) (n - 2) = 0$

Step 2

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$n = 0$

$n - 4 = 0$

$n - 2 = 0$

Step 3

Set

$n$ equal to

$0$ .

$n = 0$

Step 4

Set

$n - 4$ equal to

$0$ and solve for

$n$ .

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Step 4.1

Set

$n - 4$ equal to

$0$ .

$n - 4 = 0$

Step 4.2

Add

$4$ to both sides of the equation.

$n = 4$

$n = 4$

Step 5

Set

$n - 2$ equal to

$0$ and solve for

$n$ .

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Step 5.1

Set

$n - 2$ equal to

$0$ .

$n - 2 = 0$

Step 5.2

Add

$2$ to both sides of the equation.

$n = 2$

$n = 2$

Step 6

The final solution is all the values that make

$5 n (n - 4) (n - 2) = 0$ true.

$n = 0, 4, 2$

$5 n^{3} - 30 n^{2} + 40 n = 0$

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$