Quadratics Cheat Sheet
This document was converted from Google Docs to Microsoft Word Format with XML Format (.docx)(not to be confused with .doc which is Office 97-2003 format) and then to unfiltered HTML (.html). It does look pretty good but there are always things to note when you're working off something like this. Firstly, the source for this is a HTML document instead of GFM so it's not human-readable. Secondly, `jekyll build` seems to not like the Therefore sign and has changed anything in a container with it to Times New Roman. I'm not sure how this actually translates on the remote repo but that's what I'm getting locally. The Google Docs version of this is much better anyway:
© Adrian Widjaja 2019 https://www.youtube.com/watch?v=v9ZlVr38HbE - https://www.youtube.com/watch?v=gIuMU8UBxR4
Expand: Binomial Products
FORMULA | RULE | EXAMPLE |
(a+b)(c+d) = ac+ad+bc+bd | Use FOIL | (3x + 5)(2x + 3) = 6x²+ 6x + 10x + 15 = 6x² + 16x + 16 |
Expand: Perfect Squares
FORMULA | RULE | EXAMPLE |
(a+b)^{2} or (a+b)(a+b) = a^{2}+2ab+b^{2} | Square the first 2x the product of the middle Square the last | (2x - 3)² = 4x² - 12x + 9 |
Expand: Difference of Two Squares
FORMULA | RULE | EXAMPLE |
(a+b)(a-b) = a²–b² ^{} | Square the first minus square the last | (3x + 2y)(3x - 2y) = 9x² - 4y² |
Factor: Difference of Two Squares
FORMULA | RULE | EXAMPLE |
a²-b² = (a+b)(a-b)^{} | Find the root of b term and solve as shown in the formula | x² - 9 = (x+3)(x-3) |
Quick Note: Factoring
FORMULA | EXPLANATION | EXAMPLE |
a(b-c) Also consider: a(b-c) = -a(c-b) = -a(-b+c) | Essentially is the inverse operation of the brackets, flipping the terms around. Used to help simplify. | x(x-5) - 2(5-x) = x(x-5) - 2(-x+5) = x(x-5) + 2(x-5) = (x-5)(x+2) |
Quick Note: Difference of Two Squares
FORMULA | RULE | EXAMPLE |
x²–(a+b)² = (x-(a+b))(x+(a+b)) | NA | 4–(x+2)² = (2–(x+2))(2+(x+2)) = (-x)(2+x+2) = -x(x+4) |
Factor: Monic Trinomials
FORMULA | RULE | EXAMPLE |
Where p and q are factors of c that add to make b x²+bx+c = (x+p)(x+q) Note: px + qx = bx ∴ p + q = b pq = c | Find two integers that multiply to c and add to b then write it in the form (x+p)(x+q) where p and q are the two numbers. | x²+9x+20 =(x+4)(x+5) |
Expand into a Monic Trinomial
FORMULA | RULE | EXAMPLE |
(x+a)(x+b) = x²+ax+bx+ab = x²+bx+c | Use FOIL Factorise the middle terms | (x+5)(x+3) = x²+3x+5x+15 = x²+x(3+5)+15 = x²+8x+15 |
Grouping in Pairs
FORMULA | RULE | EXAMPLE |
Where p and q are integers that add to ab and multiply to c: x²+ax+bx+c | Group the 4 terms into pairs and take out the common binomial factor *Also read Expand into a Monic Trinomial | x²+4x+3x+12 = x(x+4)+3(x+4) = (x+4)(x+3) |
Factor: Non-Monic Trinomials
FORMULA | RULE | EXAMPLE |
Where p and q are integers that add to b and multiply to ac ax²+bx+c | Non-monic trinomial: ● Coefficient of x² is not 1 ● No common factors between the terms Look for factors of ac that add to b | 2x2+3x+2 = 2x2+1x+2x+2 = (2x+1)(x+2) |
Solving Quadratics
FORMULA | RULE/EXPLANATION | EXAMPLE |
(x+p)(x+q) | The Null Factor Law states: If ab=0, then a=0 or b=0. | (x+4)(x-9) ∴x = -5 or +9 |
Quick Note: Indices
I know that this cheat sheet is Quadratics but I also forget a few indices things so I’m putting this here anyway and you can’t do anything about it.
RULE/EXPLANATION | EXAMPLE |
Multiplying: Keep the base, add the powers Power to a power: Keep the base, multiply the powers | x²x x³= x^{5} x^{6} ÷ x^{4} = x^{2} (x^{2})^{3} = x^{6} x^{-9} = 1/x^{9} x^{1/2} = √x x^{3/5} = ^{5}√x^{3} |