BSB Year 9 is working well at factorisation (we called it factorising in lessons; American and Canadian spelling is factorizing) and this page is written for them.
Theory: Using the first four greek letters, α,β,γ,δ;
(αx+β)(γx+δ) = αγx2 + (αδ+βγ)x +βδ so for any quadratic expression ax2 + bx + c then ac=αβγδ that is a and c multiply to the number αβγδ which has alternative factors αδ and βγ. this leads to the understanding that the class has: if the x2 coefficient is not one then multiply the two end numbers and look for factors of that number [αβγδ] that add to make the middle (x) coefficient.
Factorise these expressions: Note that some questions are connected. Assume that if a homework is set form this page then it is intended that you do questions 1-20 OR 21-40 OR 41-50.
1 x² + 8x + 12 21 x² + 8x + 12 41 x² + 8x + 12
2 x² + 8x + 7 22 x² + 4x - 12 42 x² + 8x + 12
3 x² + 8x +15 23 x² + 13x + 12 43 x² + 8x + 12
4 3x² + 16x + 5 24 x² - x - 12 44 x² + 8x + 12
5 5x² - 8x + 3 25 2x² + x - 6 45 x² + 8x + 12
6 2x²+ 16x + 24 26 x² + 8x + 24 46 x² + 8x + 12
7 2x² + 5x - 12 27 2x² + 8x + 12 47 x² + 8x + 12
8 2x² - 10x - 12 28 3x² + 8x + 8 48 x² + 8x + 12
9 12x² - 10x + 2 29 x² + 8x + 12 49 x² + 8x + 12
10 12x² - 5x - 2 30 x² + 8x + 12 50 x² + 8x + 12
Be careful to factorise a question completely. If a term has a common factor left in it, e.g. (2x+8) then you have not quite finished, as 2x+8 = 2(x+4). Good students spot a common factor in all three terms before they start writing a pair of brackets. The best students check both before and afterwards - but the issue is, did you succeed in factorising completely?.
Solving quadratics starts off with factorisation. When the factorisation is finished there is a result of the form (αx+β)(γx+δ) = 0. This is two terms (or numbers, for a value of x) which multiply to make zero. So one of them is zero. Some people read this but do not ‘get’ it; make sure you are not among those folk.
As x changes values, there will be two solutions, one for each term. Either αx+β = 0 [which means x = -β/α] or γx+δ = 0 [which would mean that x = -δ/γ]. Do not confuse the process of finding the value of x for which a term is zero with the factorisation process. Solving produces roots of the equation; factorising only finds factors.
Solve these.
11 x² + 8x + 12 = 0 (both roots are negative)
12 x² - 8x + 15 = 0 (both roots are positive)
13 x² - 18x + 45 = 0
14 3x² + 16x + 5 = 0
15 5x² - 8x + 3 = 0
....now write out all the factors of 144 (yes, really) and then attempt these harder questions:
16 x² + 15x + 144 = 0
17 2x² + 18x - 72 = 0
18 4x² - 10x - 36 = 0
19 12x² - 51x + 12 = 0
20 24x² - 7x - 6 = 0
Again, I may well extend these by another ten or twenty problems.
DJS 20130415
