Question

How do you factor `7x^2 - 9x - 10`?

Answer 1

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

Explanation:

Let's start by finding the zeroes of the trynomial:

`x=(9+-sqrt(81-4(7)(-10)))/(2(7))`

`x=(9+-sqrt(81+280))/4`

`x=(9+-sqrt(361))/14`

`x=(9+-19)/14`

`x=-10/14 and x=28/14`

that's

`x_1=-5/7 and x_2=2`

Since a trynomial can be factorized by the formula:

`ax^2+bx+c=a(x-x_1)(x-x_2)`

so

`7x^2-9x+10=7(x+5/7)(x-2)`

or

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

Answer 2

(7x + 5)(x - 2)

Explanation:

Use the new AC method to factor trinomials (Socratic Search)
`y = 7x^2 - 9x - 10 =` 7(x + p)(x + q)
Converted trinomial: `y' = x^2 - 9x - 70 =` (x + p')(x + q').
Find p' ans q' knowing they have opposite sign since ac < 0.
Factor pairs of (ac = -70) --> ...(-5, 14)(5, - 14). This sum is (-9 = b). Then, p' = 5 and q' = -14.
Back to original y --> `p = (p')/a = 5/7`, and `q = (q')/a = -14/7 = -2`.
Factored form: `y = 7(x + 5/7)(x - 2) = (7x + 5)(x - 2)`