Pre-Algebra Examples

$66 = \frac{t^{2} - t}{2}$

Step 1

Rewrite the equation as $\frac{t^{2} - t}{2} = 66$ .

$\frac{t^{2} - t}{2} = 66$

Step 2

Multiply both sides of the equation by $2$ .

$2 \frac{t^{2} - t}{2} = 2 \cdot 66$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $2$ .

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Step 3.1.1.1

Cancel the common factor.

$2 \frac{t^{2} - t}{2} = 2 \cdot 66$

Step 3.1.1.2

Rewrite the expression.

$t^{2} - t = 2 \cdot 66$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $66$ .

$t^{2} - t = 132$

Step 4

Subtract $132$ from both sides of the equation.

$t^{2} - t - 132 = 0$

Step 5

Factor $t^{2} - t - 132$ using the AC method.

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Step 5.1

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 $- 132$ and whose sum is $- 1$ .

$- 12, 11$

Step 5.2

Write the factored form using these integers.

$(t - 12) (t + 11) = 0$

Step 6

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

$t - 12 = 0$

$t + 11 = 0$

Step 7

Set $t - 12$ equal to $0$ and solve for $t$ .

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Step 7.1

Set $t - 12$ equal to $0$ .

$t - 12 = 0$

Step 7.2

Add $12$ to both sides of the equation.

$t = 12$

Step 8

Set $t + 11$ equal to $0$ and solve for $t$ .

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Step 8.1

Set $t + 11$ equal to $0$ .

$t + 11 = 0$

Step 8.2

Subtract $11$ from both sides of the equation.

$t = - 11$

Step 9

The final solution is all the values that make $(t - 12) (t + 11) = 0$ true.

$t = 12, - 11$