# Further Quadratic Expressions

GCSE(H),

Factorising the expression 9x2 - 25x - 6 includes a coefficient for the x2 term (9). The factorised expression will therefore be of the form (px + q)(rx + s), where p, q, r and s are integers.

There are three known facts about the factors:

p x r = 9; therefore p and r are factors of 9;

q x s = -6; therefore q and s are factors of -6;

ps + qr = -25.

The factors of 9 are (1, 9) and (3, 3). The factors of -6 are (1, -6), (2, -3), (3, -2), (6, -1), (-1, 6), (-2, 3), (-3, 2), (-6, 1). The answer involves a combination of these factor sets: often they can be determined by inspection; that is, looking for an immediate answer.

The expression factorises to (9x + 2)(x - 3).

## Examples

1. Factorise 8x2 + 22x + 5.

Answer: (4x + 1)(2x + 5)

The factors of 5 are (1, 5). The factors of 8 are (1, 8), (2, 4), (4, 2). Set out a table to determine which combinations of these factors give 22.

The coefficients of x are 2 and 2; the integers are 2 and 3.

2. Factorise -21x2 + 29x - 10.

Answer: (-7x + 5)(3x - 2)

Look at the positive factors only for 21: (1, 21), (3, 7), and 10: (1, 10), (2, 5), to see which combination gives an answer of 29:

 (1, 10) (2, 5) (1, 21) 1x21+10x1=31 2x21+5x1=47 (3, 7) 1x7+10x3=37 2x7+5x3=29

The combination is based on (7x + 5)(3x + 2); check positive/negative combinations to obtain (-7x + 5)(3x - 2)