Factorising Trinomials

[vague oxide]

How do I do i factorise these questions

[golden cloud]

Find two numbers $p$ and $q$ such that:
$$p + q = b$$
$$p \times q = a \times c$$
Split your trinomial:
$$ax^2 + px + qx + c$$
Group each pair:
$$\qty(ax^2 + px) + (qx + c)$$
Rest is trivial

[worn spruce]

King Leo's Teacup

[static cove]

(x+3)(x+1)

[golden cloud]

✅

[bright flame]

there are 2 methods i would like to introduce

the first one is the good old completing squares

so if you have x^2 + 4x + 3 here

you need to try to make (x + 2)^2 (since we have x^2 + 4x)

(x + 2)^2 = x^2 + 4x + 4

so the expression should become x^2 + 4x + 4 - 1

or (x + 2)^2 - 1

and according to the difference of squares formula

the solution should be (x + 1)(x + 3)

and the second way

i often use it

for every second degree polynomial f(x) = ax^2 + bx + c with |a| > 0, you need to split b into b1 and b2 such that these 2 multiply into ac

so for your example

x^2 + 4x + 3 = x^2 + x + 3x + 3 (because 1 * 3 = 1 * 3 and 1 + 3 = 4)

so now you factor

like this:

x(x + 1) + 3(x + 1)