Equation and solution:
15x² - 108x + 108
15x² - 90x - 18x + 108
The following lines are not relevant to the question and are included only for completion:
15x( x - 6 ) - 18( x - 6 )
( 15x - 18 )( x - 6 )
3( 5x - 6 )( x - 6 )
My question is:
To find that we have to use - 90x - 18x in place of -108x my only known approach is trial and error.
I am wondering if there is any method for calculating these two numbers or narrowing it down as larger numbers can take quite a few attempts before finding the correct ones.
+118654 15x² - 108x + 108
This is what I would do.
First take out common factors I mean take out the 3
\(=3(5x^2-36x+36)\)
Now I want two numbers that multiply to be 5*36=180
Since the multiple of 5 and 36 is positive the two numbers must both be positive OR both be negative.
The numbers need to add to -36 (the middle one) so they must both be negative.
Here I do use a bit of trial and error
5*36=5*2*3*2*3 = 5*6*6 = 30*6 bingo the numbers are -30 and -6
-30*-6=180
-30+-6=-36
\(=3(5x-30x-6x+36)\)
Anyway, I hope that helps ;)
Another way to do it is to utilize the quadratic formula from that you can just gives the final factors not the break down of it.
\(\)
Factor the following:
15 x^2 - 108 x + 108
Factor 3 out of 15 x^2 - 108 x + 108:
3 (5 x^2 - 36 x + 36)
Factor the quadratic 5 x^2 - 36 x + 36. The coefficient of x^2 is 5 and the constant term is 36. The product of 5 and 36 is 180. The factors of 180 which sum to -36 are -6 and -30. So 5 x^2 - 36 x + 36 = 5 x^2 - 30 x - 6 x + 36 = x (5 x - 6) - 6 (5 x - 6):
3 x (5 x - 6) - 6 (5 x - 6)
Factor 5 x - 6 from x (5 x - 6) - 6 (5 x - 6):
Answer: |3 (5 x - 6) (x - 6)
