Cramer's rules + Po Shen Loh

Question 1

Cramer’s rule: one linear equation with a single variable

Answer 1

Question 2

Cramer’s rule: system of two linear equations with two variables

Answer 2

Question 3

Po Shen Loh method for quadratic equations

Answer 3

Example: r + s = 8, rs = 13

Two numbers add up to 8 when and only when their average is equal to 4 (we divide 8 by 2). And two numbers will have an average of 4 when and only when these two numbers are (4-e) and (4+e), where e is a step in either direction from their average which is 4.

At this point, we will use the information about the product of the roots:
(4 – e) x (4 + e) = 13
Let’s open the brackets: 16 - e^2 = 13, hence e^2 = 3, hence e = +√3 or -√3, we can use either, let’s use +√3.

Hence, the roots of this quadratic equation are: 4 -√3 and 4 +√3.
Note: if we use -√3 instead, we will get again: 4 +√3 and 4 -√3.

The equation will be X^2 -8X + 13 = 0