Finite Math Examples

$12 - \frac{3}{8} (2 t^{2} - 26 t + 89)$

Step 1

Apply the distributive property.

$12 - \frac{3}{8} (2 t^{2}) - \frac{3}{8} (- 26 t) - \frac{3}{8} \cdot 89$

Step 2

Simplify.

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Step 2.1

Cancel the common factor of $2$ .

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Step 2.1.1

Move the leading negative in $- \frac{3}{8}$ into the numerator.

$12 + \frac{- 3}{8} (2 t^{2}) - \frac{3}{8} (- 26 t) - \frac{3}{8} \cdot 89$

Step 2.1.2

Factor $2$ out of $8$ .

$12 + \frac{- 3}{2 (4)} (2 t^{2}) - \frac{3}{8} (- 26 t) - \frac{3}{8} \cdot 89$

Step 2.1.3

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

$12 + \frac{- 3}{2 (4)} (2 (t^{2})) - \frac{3}{8} (- 26 t) - \frac{3}{8} \cdot 89$

Step 2.1.4

Cancel the common factor.

$12 + \frac{- 3}{2 \cdot 4} (2 t^{2}) - \frac{3}{8} (- 26 t) - \frac{3}{8} \cdot 89$

Step 2.1.5

Rewrite the expression.

$12 + \frac{- 3}{4} t^{2} - \frac{3}{8} (- 26 t) - \frac{3}{8} \cdot 89$

Step 2.2

Combine $\frac{- 3}{4}$ and $t^{2}$ .

$12 + \frac{- 3 t^{2}}{4} - \frac{3}{8} (- 26 t) - \frac{3}{8} \cdot 89$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Move the leading negative in $- \frac{3}{8}$ into the numerator.

$12 + \frac{- 3 t^{2}}{4} + \frac{- 3}{8} (- 26 t) - \frac{3}{8} \cdot 89$

Step 2.3.2

Factor $2$ out of $8$ .

$12 + \frac{- 3 t^{2}}{4} + \frac{- 3}{2 (4)} (- 26 t) - \frac{3}{8} \cdot 89$

Step 2.3.3

Factor $2$ out of $- 26 t$ .

$12 + \frac{- 3 t^{2}}{4} + \frac{- 3}{2 (4)} (2 (- 13 t)) - \frac{3}{8} \cdot 89$

Step 2.3.4

Cancel the common factor.

$12 + \frac{- 3 t^{2}}{4} + \frac{- 3}{2 \cdot 4} (2 (- 13 t)) - \frac{3}{8} \cdot 89$

Step 2.3.5

Rewrite the expression.

$12 + \frac{- 3 t^{2}}{4} + \frac{- 3}{4} (- 13 t) - \frac{3}{8} \cdot 89$

Step 2.4

Combine $- 13$ and $\frac{- 3}{4}$ .

$12 + \frac{- 3 t^{2}}{4} + \frac{- 13 \cdot - 3}{4} t - \frac{3}{8} \cdot 89$

Step 2.5

Multiply $- 13$ by $- 3$ .

$12 + \frac{- 3 t^{2}}{4} + \frac{39}{4} t - \frac{3}{8} \cdot 89$

Step 2.6

Combine $\frac{39}{4}$ and $t$ .

$12 + \frac{- 3 t^{2}}{4} + \frac{39 t}{4} - \frac{3}{8} \cdot 89$

Step 2.7

Multiply $- \frac{3}{8} \cdot 89$ .

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Step 2.7.1

Multiply $89$ by $- 1$ .

$12 + \frac{- 3 t^{2}}{4} + \frac{39 t}{4} - 89 (\frac{3}{8})$

Step 2.7.2

Combine $- 89$ and $\frac{3}{8}$ .

$12 + \frac{- 3 t^{2}}{4} + \frac{39 t}{4} + \frac{- 89 \cdot 3}{8}$

Step 2.7.3

Multiply $- 89$ by $3$ .

$12 + \frac{- 3 t^{2}}{4} + \frac{39 t}{4} + \frac{- 267}{8}$

Step 3

Simplify each term.

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Step 3.1

Move the negative in front of the fraction.

$12 - \frac{3 t^{2}}{4} + \frac{39 t}{4} + \frac{- 267}{8}$

Step 3.2

Move the negative in front of the fraction.

$12 - \frac{3 t^{2}}{4} + \frac{39 t}{4} - \frac{267}{8}$

Step 4

To write $12$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{3 t^{2}}{4} + \frac{39 t}{4} + 12 \cdot \frac{8}{8} - \frac{267}{8}$

Step 5

Combine $12$ and $\frac{8}{8}$ .

$- \frac{3 t^{2}}{4} + \frac{39 t}{4} + \frac{12 \cdot 8}{8} - \frac{267}{8}$

Step 6

Combine the numerators over the common denominator.

$- \frac{3 t^{2}}{4} + \frac{39 t}{4} + \frac{12 \cdot 8 - 267}{8}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $12$ by $8$ .

$- \frac{3 t^{2}}{4} + \frac{39 t}{4} + \frac{96 - 267}{8}$

Step 7.2

Subtract $267$ from $96$ .

$- \frac{3 t^{2}}{4} + \frac{39 t}{4} + \frac{- 171}{8}$

Step 8

Move the negative in front of the fraction.

$- \frac{3 t^{2}}{4} + \frac{39 t}{4} - \frac{171}{8}$

Step 9

Use the quadratic formula to find the roots for $- \frac{3 t^{2}}{4} + \frac{39 t}{4} - \frac{171}{8} = 0$

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Step 9.1

Multiply through by the least common denominator $8$ , then simplify.

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Step 9.1.1

Apply the distributive property.

$8 (- \frac{3 t^{2}}{4}) + 8 (\frac{39 t}{4}) + 8 (- \frac{171}{8}) = 0$

Step 9.1.2

Simplify.

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Step 9.1.2.1

Cancel the common factor of $4$ .

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Step 9.1.2.1.1

Move the leading negative in $- \frac{3 t^{2}}{4}$ into the numerator.

$8 (\frac{- 3 t^{2}}{4}) + 8 (\frac{39 t}{4}) + 8 (- \frac{171}{8}) = 0$

Step 9.1.2.1.2

Factor $4$ out of $8$ .

$4 (2) (\frac{- 3 t^{2}}{4}) + 8 (\frac{39 t}{4}) + 8 (- \frac{171}{8}) = 0$

Step 9.1.2.1.3

Cancel the common factor.

$4 \cdot (2 (\frac{- 3 t^{2}}{4})) + 8 (\frac{39 t}{4}) + 8 (- \frac{171}{8}) = 0$

Step 9.1.2.1.4

Rewrite the expression.

$2 (- 3 t^{2}) + 8 (\frac{39 t}{4}) + 8 (- \frac{171}{8}) = 0$

Step 9.1.2.2

Multiply $- 3$ by $2$ .

$- 6 t^{2} + 8 (\frac{39 t}{4}) + 8 (- \frac{171}{8}) = 0$

Step 9.1.2.3

Cancel the common factor of $4$ .

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Step 9.1.2.3.1

Factor $4$ out of $8$ .

$- 6 t^{2} + 4 (2) (\frac{39 t}{4}) + 8 (- \frac{171}{8}) = 0$

Step 9.1.2.3.2

Cancel the common factor.

$- 6 t^{2} + 4 \cdot (2 (\frac{39 t}{4})) + 8 (- \frac{171}{8}) = 0$

Step 9.1.2.3.3

Rewrite the expression.

$- 6 t^{2} + 2 (39 t) + 8 (- \frac{171}{8}) = 0$

Step 9.1.2.4

Multiply $39$ by $2$ .

$- 6 t^{2} + 78 t + 8 (- \frac{171}{8}) = 0$

Step 9.1.2.5

Cancel the common factor of $8$ .

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Step 9.1.2.5.1

Move the leading negative in $- \frac{171}{8}$ into the numerator.

$- 6 t^{2} + 78 t + 8 (\frac{- 171}{8}) = 0$

Step 9.1.2.5.2

Cancel the common factor.

$- 6 t^{2} + 78 t + 8 (\frac{- 171}{8}) = 0$

Step 9.1.2.5.3

Rewrite the expression.

$- 6 t^{2} + 78 t - 171 = 0$

Step 9.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 9.3

Substitute the values $a = - 6$ , $b = 78$ , and $c = - 171$ into the quadratic formula and solve for $t$ .

$\frac{- 78 \pm \sqrt{78^{2} - 4 \cdot (- 6 \cdot - 171)}}{2 \cdot - 6}$

Step 9.4

Simplify.

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Step 9.4.1

Simplify the numerator.

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Step 9.4.1.1

Raise $78$ to the power of $2$ .

$t = \frac{- 78 \pm \sqrt{6084 - 4 \cdot - 6 \cdot - 171}}{2 \cdot - 6}$

Step 9.4.1.2

Multiply $- 4 \cdot - 6 \cdot - 171$ .

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Step 9.4.1.2.1

Multiply $- 4$ by $- 6$ .

$t = \frac{- 78 \pm \sqrt{6084 + 24 \cdot - 171}}{2 \cdot - 6}$

Step 9.4.1.2.2

Multiply $24$ by $- 171$ .

$t = \frac{- 78 \pm \sqrt{6084 - 4104}}{2 \cdot - 6}$

Step 9.4.1.3

Subtract $4104$ from $6084$ .

$t = \frac{- 78 \pm \sqrt{1980}}{2 \cdot - 6}$

Step 9.4.1.4

Rewrite $1980$ as $6^{2} \cdot 55$ .

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Step 9.4.1.4.1

Factor $36$ out of $1980$ .

$t = \frac{- 78 \pm \sqrt{36 (55)}}{2 \cdot - 6}$

Step 9.4.1.4.2

Rewrite $36$ as $6^{2}$ .

$t = \frac{- 78 \pm \sqrt{6^{2} \cdot 55}}{2 \cdot - 6}$

Step 9.4.1.5

Pull terms out from under the radical.

$t = \frac{- 78 \pm 6 \sqrt{55}}{2 \cdot - 6}$

Step 9.4.2

Multiply $2$ by $- 6$ .

$t = \frac{- 78 \pm 6 \sqrt{55}}{- 12}$

Step 9.4.3

Simplify $\frac{- 78 \pm 6 \sqrt{55}}{- 12}$ .

$t = \frac{13 \pm \sqrt{55}}{2}$

Step 9.5

The final answer is the combination of both solutions.

$t = \frac{13 + \sqrt{55}}{2}, \frac{13 - \sqrt{55}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$t = \frac{13 + \sqrt{55}}{2}, \frac{13 - \sqrt{55}}{2}$

Decimal Form:

$t = 10.20809924 \dots, 2.79190075 \dots$