Solve for the unknown: 4.1.1 2x2 − 14x + 12 = 0 4.1.2 y2(y−3) + 81(3−y) = 0 4.1.3 x/5 - 3x-2/3 = -1 4.1.4 A mother is currently nine times as old as her son. In four years' time, she will be five times as old as her son. What are their present ages? 4.1.5 32x = 243 4.1.6 5^x+2 = 1/625

The values of $x$ are $\boxed{1 \text{ or } 6}$.

4.1.2

The values of $y$ are $\boxed{3, 9, \text{ or } -9}$.

4.1.3

The common denominator for 5 and 3 is 15.

The value of $x$ is $\boxed{\frac{25}{12}}$.

4.1.4

Let $M$ be the mother's current age and $S$ be the son's current age.
"A mother is currently nine times as old as her son":

"In four years' time, she will be five times as old as her son":

The mother's current age is $\boxed{36 \text{ years}}$ and the son's current age is $\boxed{4 \text{ years}}$.

4.1.5

The value of $x$ is $\boxed{\frac{5}{2}}$.

4.1.6

Since $625 = 5^4$, we have:

The value of $x$ is $\boxed{-6}$.

Question 5

5.1.1

The given pattern is $9; 15; 21; 27; ...$
The difference between consecutive terms is $15-9=6$, $21-15=6$, $27-21=6$.
The common difference is 6.

The next three terms are $\boxed{33, 39, 45}$.

5.1.2
The pattern is an arithmetic sequence where each term is obtained by adding 6 to the previous term. The first term is 9.

5.1.3

Here, the first term $a=9$, the common difference $d=6$, and we want the $20^{th}$ term, so $n=20$.

The $20^{th}$ term of the pattern is $\boxed{123}$.

5.2

The given pattern is $x+2; x+3; x+4; ...$
The first term $a = x+2$.
The common difference $d = (x+3) - (x+2) = 1$.

The general rule for the pattern is $\boxed{T_n = x+n+1}$.