Complete

Quadratics Cheat Sheet

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© 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)

= a²+2ab+b²

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)
= -a(c-b)

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))
= (-a)(x+a+b)

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+x(a+b)+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
= x(x+p)+q(x+p)
= (x+p)(x+q)

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
= ax²+px+qx+c
= (ax+p)(a+q)

Non-monic trinomial:

● Coefficient of x² is not 1

● No common factors between the terms

Look for factors of ac that add to b

Split the b term

Factor in pairs

2x2+3x+2

= 2x2+1x+2x+2

= (2x+1)(x+2)

Solving Quadratics

FORMULA

RULE/EXPLANATION

EXAMPLE

(x+p)(x+q)
(x+p) = 0 or
(x+q) = 0

∴ x = -p or -q

See rule for explanation

The Null Factor Law states:

If ab=0, then a=0 or b=0.

∴ We can solve a quadratic in binomial form in stating that:

(x+5)(x-9)
∴ x = -5 or +9

(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
Dividing: Keep the base, subtract the powers

Power to a power: Keep the base, multiply the powers
Power of zero: x⁰ = 1, where x is any number

Negative indices: a^-m = 1/
Make sure that for any negative indices, you move them

Fractional indices = the root

x²x x³= x⁵

x⁶ ÷ x⁴ = x²

(x²)³ = x⁶

x^-9 = 1/x⁹

x^1/2 = √x

x^3/5 = ⁵√x³