Sure, let's tackle each question step-by-step.
### QUESTION 35: Factorize [tex]\(3t^2 - 19t - 14\)[/tex]
1. Identify the quadratic expression: We are given the quadratic expression [tex]\(3t^2 - 19t - 14\)[/tex].
2. Find two numbers that multiply to the product of the coefficient of [tex]\(t^2\)[/tex] and the constant term: This means we need numbers that multiply to [tex]\(3 \cdot (-14) = -42\)[/tex] and add up to -19.
These numbers are -21 and 2 because [tex]\((-21) \cdot 2 = -42\)[/tex] and [tex]\((-21) + 2 = -19\)[/tex].
3. Split the middle term using these factors:
Rewrite the expression as:
[tex]\[
3t^2 - 21t + 2t - 14
\][/tex]
4. Factor by grouping: Group the terms to factor by grouping.
[tex]\[
(3t^2 - 21t) + (2t - 14)
\][/tex]
Factor out the common factors in each group:
[tex]\[
3t(t - 7) + 2(t - 7)
\][/tex]
5. Factor out the common binomial factor:
Notice that [tex]\(t - 7\)[/tex] is common in both terms.
[tex]\[
(t - 7)(3t + 2)
\][/tex]
6. Write the final factorized form:
Thus, the factorized form of [tex]\(3t^2 - 19t - 14\)[/tex] is
[tex]\[
(t - 7)(3t + 2)
\][/tex]
### QUESTION 36: Calculate the mean of the data set [tex]\(\{6, 11, 14, 18, 7, 22, 6\}\)[/tex]
1. List the data points:
The data set is [tex]\(\{6, 11, 14, 18, 7, 22, 6\}\)[/tex].
2. Calculate the sum of the data points:
[tex]\[
6 + 11 + 14 + 18 + 7 + 22 + 6 = 84
\][/tex]
3. Count the number of data points:
There are 7 data points in the set.
4. Divide the sum by the number of data points to find the mean:
[tex]\[
\text{Mean} = \frac{\text{Sum of data points}}{\text{Number of data points}} = \frac{84}{7} = 12.0
\][/tex]
Thus, the mean of the data set [tex]\(\{6, 11, 14, 18, 7, 22, 6\}\)[/tex] is [tex]\(12.0\)[/tex].
