Question

How do you factor `10x^2 - 19x + 7 = 0`?

Answer 1

`(5x -7)(2x -1) =0`

Explanation:

Find factors of 10 an 7 which add up to 19.
The signs will be the same, both minus.

7 is a prime number so has to be 7 x 1.

10 can be 10 x 1 or 5 x 2.
Cross multiply and add

`" 5 7" rArr 2 xx7 = 14`
`" 2 1" rArr 5 xx1 = 5`
`" "14+5=19`

The top row is the first bracket, the bottom row is the second bracket.

`(5x -7)(2x -1)`

You can always multiply out to check that the combinations are correct.

Answer 2

`(2x - 1)(5x - 7)`

Explanation:

The best way is using the new AC Method (Socratic Search)
`y = 10x^2 - 19x + 7 = 10(x + p)(x + q)`
Converted trinomial: `y' = x^2 - 19x + 70 = (x + p')(x + q')`.
p' and q 'have same sign (Rule of signs). Compose factor pairs of (ac = 70): --> (-2, -35),(-5, 1-4). This last sum is (-19 = b). Consequently,
p' = -5 and q' = -14.
Back to original trinomial y, `p = (p')/a = -5/10 = -1/2`, and
`q = (q')/a = -14/10 = -7/5`
Factored form: `y = 10(x - 1/2)(x - 7/5) = (2x - 1)(5x - 7)`