Question

How do you factor `5y^2-12y+4=0`?

Answer 1

`(5x-2)(x-2)=0`

Explanation:

Read all the clues which are given in the quadratic trinomial:

`color(turquoise)5y^2 color(orange)(-)color(magenta)(12)y color(red)(+)color(turquoise)4 = 0`

We need to find factors of `color(turquoise)(5 and 4)`

The factors of 5 and 4 must `color(red)"(ADD)"` together to give `color(magenta)(12)`

The signs in the brackets will be `color(red)"THE SAME"`
They will BOTH be `color(orange)"(NEGATIVE)"`"

Use factors of 5 and 4 and test the cross-products:

`color(white)(.....) 5 " "2 " " rArr 1xx2 = 2`
`color(white)(.....) 1 " "2" " rArr 5xx2 = 10" " 10+2 =color(magenta)(12)`

The top row gives the factors in one bracket.
The second gives the factors in the other bracket.

`(5x-2)(x-2)=0`

Solving would lead to `x = 2/5 or x = 2`

Answer 2

(5x - 2)(x - 2)

Explanation:

Use the new AC Method (Socratic Search)
`f(y) = 5y^2 - 12y + 4 = 5(x + p)(x + q)`
Converted trinomial `f'(y)' = y^2 - 12 y + 20 =` (x + p')(x + q').
Method. Find p' and q' then divide them by a to get p and q.
p' and q' have same sign because ac > 0.
Factor pairs of (ac = 20) --> (2, 10)(-2, -10). This last sum is -12 = b.
Then, p' = -2 and q' = -10
Back to f(y) --> `p = (p')/a = -2/5`. and `q = (q')/a = -10/5 = -2`.
Factor form of f(y):
`f(y) = 5(x - 2/5)(x - 2) = (5x - 2)(x - 2)`