Basic Quadratic Factorisation Revision



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Types of Quadratic

A quadratic expression is functions of the form ...

ax2+bx+c

where a, b and c are constants.

or, factorised ...

(px+q)(rx+s)

which is a factorised version of the first, where p, q, r and s are constants.

or, completed square form ...

α(x+β)+γ

again another version of the first, where α, β and γ are constants.


Fully factorise the following:

(Hint: always look for a common factor first, if available, then check if other forms of factorisation are possible)

  1. x2+8x
  2. x2+4x+4
  3. x2+4x-12
  4. x2+4x+5
  5. x4-4x3
  6. x2-12x+35
  7. z2-z-72
  8. y2+2y-120
  9. m2-144
  10. p3+5p2-36p
  11. q4-16q2+64
  12. p2q2-6pq+9

Challenge

How many integer values of a are there for which x2+ax+100 is factorisable? How about x2+ax-100?

Answers (x+2)^2 x(x+4) (x+6)(x-2) (x+4)(x+1) x^3 (x-4) (x-7)(x-5) (x+8)(x-9) (y+12)(y-10) p(p+9)(p-4) (p^2+7)^2 (pq-3)^2 Note that 100=100×1=50×2=25×4=20×5=10×10=-100×-1=⋯ So (x+100)(x+1) would be a factorisation and so on, giving 10 factorisations. If 100 is negative, then there are more factorisations because 20×-5 is distinct from -20×5. That means we nearly have double then. However (x+10)(x-10) is the same as (x-10)(x+10), so it’s 1 less than double, i.e. 19 factorisations.

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