Finite Math Examples

$t^{4} - 4 t^{3} - 32 t^{2} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $t^{2}$ out of $t^{4} - 4 t^{3} - 32 t^{2}$ .

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Step 1.1.1

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

$t^{2} t^{2} - 4 t^{3} - 32 t^{2} = 0$

Step 1.1.2

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

$t^{2} t^{2} + t^{2} (- 4 t) - 32 t^{2} = 0$

Step 1.1.3

Factor $t^{2}$ out of $- 32 t^{2}$ .

$t^{2} t^{2} + t^{2} (- 4 t) + t^{2} \cdot - 32 = 0$

Step 1.1.4

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

$t^{2} (t^{2} - 4 t) + t^{2} \cdot - 32 = 0$

Step 1.1.5

Factor $t^{2}$ out of $t^{2} (t^{2} - 4 t) + t^{2} \cdot - 32$ .

$t^{2} (t^{2} - 4 t - 32) = 0$

Step 1.2

Factor.

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Step 1.2.1

Factor $t^{2} - 4 t - 32$ 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 $- 32$ and whose sum is $- 4$ .

$- 8, 4$

Step 1.2.1.2

Write the factored form using these integers.

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

Step 1.2.2

Remove unnecessary parentheses.

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

Step 2

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

$t^{2} = 0$

$t - 8 = 0$

$t + 4 = 0$

Step 3

Set $t^{2}$ equal to $0$ and solve for $t$ .

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Step 3.1

Set $t^{2}$ equal to $0$ .

$t^{2} = 0$

Step 3.2

Solve $t^{2} = 0$ for $t$ .

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Step 3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{2}$ .

$t = \pm \sqrt{0^{2}}$

Step 3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$t = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$t = 0$

Step 4

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

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Step 4.1

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

$t - 8 = 0$

Step 4.2

Add $8$ to both sides of the equation.

$t = 8$

Step 5

Set $t + 4$ equal to $0$ and solve for $t$ .

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Step 5.1

Set $t + 4$ equal to $0$ .

$t + 4 = 0$

Step 5.2

Subtract $4$ from both sides of the equation.

$t = - 4$

Step 6

The final solution is all the values that make $t^{2} (t - 8) (t + 4) = 0$ true.

$t = 0, 8, - 4$