Question

How do you factor completely: `35x^2-54x+16`?

Answer 1

Use quadratic formula to find roots of `35x^2-54x+16 = 0` and deduce:

`35x^2-54x+16 = (5x-2)(7x-8)`

Explanation:

Let `f(x) = 35x^2-54x+16`

This is of the form `ax^2+bx+c`, with `a=35`, `b=-54` and `c=16`.

`f(x) = 0` has roots given by the quadratic formula:

`x = (-b+-sqrt(b^2-4ac))/(2a)`

`=(54+-sqrt((-54)^2-(4xx35xx16)))/(2xx35)`

`=(54+-sqrt(2916 - 2240))/70`

`=(54+-sqrt(676))/70`

`=(54+-26)/70`

That is `x = 28/70 = 2/5` or `x = 80/70 = 8/7`

So `(5x-2) = 0` or `(7x-8) = 0`

Hence:

`35x^2-54x+16 = (5x-2)(7x-8)`

Answer 2

Factor `y = 35x^2 - 54x + 16`

Ans: `(5x - 2)(7x - 8)`

Explanation:

I use the new AC Method to factor trinomials (Google, Yahoo Search)

`y = 35x^2 - 54x + 16 =`35(x + p)(x + q)

Converted trinomial: `y' = x^2 - 54x + 560.`
Find 2 numbers p' and q' knowing sum (-54 = b) and product (ac = 560)'. Here p' and q' have same sign.
Factor pairs of ac = 560 --> (10, 56)(14, 40). This sum is 54 = -b. Change the sum to the opposite.
Then p' = -14, and q' = -40, Therefore,
`p = (p')/a = -14/35 = -2/5` and `q = (q')/a = -40/35 = -8/7`.

Factored form:

`y = 35(x - 2/5)(x - 8/7) = (5x - 2)(7x - 8)`