Question

How do you factor `12x^2+69x+45`?

Answer 1

The equation can be factorised to `3(4x+3)(x+5)`

Explanation:

Use the general formula
`x = (-b +- sqrt(b^2 -4ac))/(2a)`

Substitute in the given values
`x = (-69 +- sqrt(69^2 -4*12*45))/(24)`

simplify
`x = (-69 +- 51)/(24)`

`x_"1" = -3/4`
`x_"2" = -5`

Put those values back into `a(x-x_"1")(x-x_"2")`

gives

`12(x +3/4)(x+5)`

rearranging
`3(4x+3)(x+5)`

Answer 2

f(x) = 3(4x + 3)(x + 5)

Explanation:

`f(x) = 3y = 12x^2 + 69x + 45 = 3(4x^2 + 23x + 15)`
Factor the trinomial y in parentheses.
`y = 4x^2 + 23x + 15`
Use the new AC Method to factor trinomials (Socratic Search)
y = 4(x + p)(x + q)
Converted trinomial:
`y' = x^2 + 23x + 60 =` (x + p')(x + q')
Find 2 numbers knowing sum (b = 23) and product (ac = 60).
They are: 3 and 20. Back to y, we get:
`p = (p')/a = 3/4`, and `q = (q')/a = 20/4 = 5`
Factored form:
y = 4(x + 3/4)(x + 5) = (4x + 3)(x + 5)
f(x) = 3y = 3(4x + 3)(x + 5)