Finite Math Examples

$0 = - 5 t^{2} + 30 t + 80$

Step 1

Rewrite the equation as $- 5 t^{2} + 30 t + 80 = 0$ .

$- 5 t^{2} + 30 t + 80 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $- 5$ out of $- 5 t^{2} + 30 t + 80$ .

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Step 2.1.1

Factor $- 5$ out of $- 5 t^{2}$ .

$- 5 t^{2} + 30 t + 80 = 0$

Step 2.1.2

Factor $- 5$ out of $30 t$ .

$- 5 t^{2} - 5 (- 6 t) + 80 = 0$

Step 2.1.3

Factor $- 5$ out of $80$ .

$- 5 t^{2} - 5 (- 6 t) - 5 \cdot - 16 = 0$

Step 2.1.4

Factor $- 5$ out of $- 5 (t^{2}) - 5 (- 6 t)$ .

$- 5 (t^{2} - 6 t) - 5 \cdot - 16 = 0$

Step 2.1.5

Factor $- 5$ out of $- 5 (t^{2} - 6 t) - 5 (- 16)$ .

$- 5 (t^{2} - 6 t - 16) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $t^{2} - 6 t - 16$ using the AC method.

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Step 2.2.1.1

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 $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Step 2.2.1.2

Write the factored form using these integers.

$- 5 ((t - 8) (t + 2)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$- 5 (t - 8) (t + 2) = 0$

Step 3

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

$t - 8 = 0$

$t + 2 = 0$

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 $t + 2$ equal to $0$ and solve for $t$ .

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Step 5.1

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

$t + 2 = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$t = - 2$

Step 6

The final solution is all the values that make $- 5 (t - 8) (t + 2) = 0$ true.

$t = 8, - 2$