Calculus Examples

$6 = \frac{t^{2} - t}{2}$

Rewrite the equation as $\frac{t^{2} - t}{2} = 6$ .

$\frac{t^{2} - t}{2} = 6$

Multiply both sides of the equation by $2$ .

$2 \frac{t^{2} - t}{2} = 2 \cdot 6$

Simplify both sides of the equation.

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Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$2 \frac{t^{2} - t}{2} = 2 \cdot 6$

Rewrite the expression.

$t^{2} - t = 2 \cdot 6$

Simplify the right side.

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Multiply $2$ by $6$ .

$t^{2} - t = 12$

Subtract $12$ from both sides of the equation.

$t^{2} - t - 12 = 0$

Factor $t^{2} - t - 12$ using the AC method.

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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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Write the factored form using these integers.

$(t - 4) (t + 3) = 0$

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

$t - 4 = 0$

$t + 3 = 0$

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

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

$t - 4 = 0$

Add $4$ to both sides of the equation.

$t = 4$

Set $t + 3$ equal to $0$ and solve for $t$ .

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Set $t + 3$ equal to $0$ .

$t + 3 = 0$

Subtract $3$ from both sides of the equation.

$t = - 3$

The final solution is all the values that make $(t - 4) (t + 3) = 0$ true.

$t = 4, - 3$