If you know that the quadratic is going to factorise, then you need to look at the coefficient of the x^2 term. The number of x^2 you have can be split into factors. For example, if the coefficient is 6, then we have pairs of factors (1,6),(2,3). Then we look at the constant term and split that into possible pairs of factors. For example, if the constant is 24 then the pairs are (1,24),(2,12),(3,8),(4,6).
Then we look at the sign of the constant. If it's + then in the next step we are going to ADD, otherwise we're going to SUBTRACT.
The next bit is like a game of tennis doubles. We'll call the game Quadrattack. Instead of the players being in a tennis court, they're in brackets, where the contents of the brackets consist of a team of two players. There's a pair of brackets, one for each team of two. Here's what the brackets look like as factors for the quadratic: (ax+b)(cx+d). We have two teams a and b versus c and d. The team members each represent a number (which may be printed on their T-shirts).
The tables below show all possible "games" (fixtures) for the two teams a and b versus c and d, the second table being the "swapped partner" set where b and d change teams. The tables allow you to solve for a, b, c, d where:
AX^2+BX+C=(aX+b)(cX+d) and ac=A and bd=C. The specific example of 6X^2+...+24 is given so ac=6 and bd=24. The tables show all possible values for B. X is any variable (representing x, y, t, etc.). The products ad and bc are calculated by multiplying the numbers on their T-shirts. Only players with the right numbers on their T-shirts can take part in the series of games.
Plus or minus (+) in front of C tells you whether to use the Sum column (+) or Diff column (-). On with the game!
Game fixtures
| Game |
a |
b |
c (=A/a) |
d (=C/b) |
ad |
bc |
Sum |
Diff |
| 1 |
1 |
1 |
6 |
24 |
24 |
6 |
30 |
18 |
| 2 |
1 |
2 |
6 |
12 |
12 |
12 |
24 |
0 |
| 3 |
1 |
3 |
6 |
8 |
8 |
18 |
26 |
10 |
| 4 |
1 |
4 |
6 |
6 |
6 |
24 |
30 |
18 |
| 5 |
2 |
1 |
3 |
24 |
48 |
3 |
51 |
45 |
| 6 |
2 |
2 |
3 |
12 |
24 |
6 |
30 |
18 |
| 7 |
2 |
3 |
3 |
8 |
16 |
9 |
25 |
7 |
| 8 |
2 |
4 |
3 |
6 |
12 |
12 |
24 |
0 |
Swap partners (b and d)
| Game |
a |
b |
c (=A/a) |
d (=C/b) |
ad |
bc |
Sum |
Diff |
| 9 |
1 |
24 |
6 |
1 |
1 |
144 |
145 |
143 |
| 10 |
1 |
12 |
6 |
2 |
2 |
72 |
74 |
70 |
| 11 |
1 |
8 |
6 |
3 |
3 |
48 |
51 |
3 |
| 12 |
1 |
6 |
6 |
4 |
4 |
36 |
40 |
4 |
| 13 |
2 |
24 |
3 |
1 |
2 |
72 |
74 |
2 |
| 14 |
2 |
12 |
3 |
2 |
4 |
36 |
40 |
4 |
| 15 |
2 |
8 |
3 |
3 |
6 |
24 |
30 |
6 |
| 16 |
2 |
6 |
3 |
4 |
8 |
18 |
26 |
8 |
In an actual question to solve or factorise a quadratic when you know that it can be factorised into rational real factors, you would set up the tables of fixtures for the games according to the given values of A and C. I've just selected A=6 and C=24 as an example. The repeated values of Sum and Diff effectively tell you that the games are in fact identical, so in the example Games 1, 4 and 6 have the same Sum, and it would not be necessary to play more than one of them if you have +C. Whoever decides the games (devises the quadratics) should pick values of A, B and C so that no two games are the same.
In Games 2 and 8 Diff is zero, which means the quadratic contains no X term (B=0).
The purpose of these games is to play each one until the Sum (where we have +C) or Diff (where we have -C) matches the given value of B. Both teams receive the prize or points, or the individual players receive points. There's just one more thing to do after the winning game: what is the sign in each bracket? For +C the sign is minus for -B and plus for +B; for -C and -B the sign is minus for the b or d player with bc or ad as the larger number and plus for the other b or d player; for -C and +B the sign is plus for the b or d player with bc or ad as the larger number and minus for the other b or d player. For example, if the quadratic to be solved was 6x^2-10x-24, then Game 3 is the winning game (Diff=10) and the factors are (x-3)(6x+8). Here we have -C (-24) and -B (-10) and bc (3*6) is larger than ad (1*8), so b (3) is minus and d (8) is plus.