Calculus Examples

$4 t^{3} - 12 t^{2} + 8 t = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $4 t$ out of $4 t^{3} - 12 t^{2} + 8 t$ .

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Step 1.1.1

Factor $4 t$ out of $4 t^{3}$ .

$4 t (t^{2}) - 12 t^{2} + 8 t = 0$

Step 1.1.2

Factor $4 t$ out of $- 12 t^{2}$ .

$4 t (t^{2}) + 4 t (- 3 t) + 8 t = 0$

Step 1.1.3

Factor $4 t$ out of $8 t$ .

$4 t (t^{2}) + 4 t (- 3 t) + 4 t (2) = 0$

Step 1.1.4

Factor $4 t$ out of $4 t (t^{2}) + 4 t (- 3 t)$ .

$4 t (t^{2} - 3 t) + 4 t (2) = 0$

Step 1.1.5

Factor $4 t$ out of $4 t (t^{2} - 3 t) + 4 t (2)$ .

$4 t (t^{2} - 3 t + 2) = 0$

Step 1.2

Factor.

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Step 1.2.1

Factor $t^{2} - 3 t + 2$ 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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 1.2.1.2

Write the factored form using these integers.

$4 t ((t - 2) (t - 1)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$4 t (t - 2) (t - 1) = 0$

Step 2

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

$t = 0$

$t - 2 = 0$

$t - 1 = 0$

Step 3

Set $t$ equal to $0$ .

$t = 0$

Step 4

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

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Step 4.1

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

$t - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$t = 2$

Step 5

Set $t - 1$ equal to $0$ and solve for $t$ .

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Step 5.1

Set $t - 1$ equal to $0$ .

$t - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$t = 1$

Step 6

The final solution is all the values that make $4 t (t - 2) (t - 1) = 0$ true.

$t = 0, 2, 1$