Question

How do you factor `5x^2-17x+14`?

Answer 1

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

Explanation:

First you would find the zeros `x_1,x_2` of the polynomial `5x^2-17x+14`

You would apply the quadratic formula:

`x_(1,2)=(17+-sqrt(17^2-4*5*14))/(2*5)`

`=(17+-sqrt(289-280))/10=(17+-3)/10`

`x_1=14/10=7/5` and `x_2=20/10=2`

Then you would factor the given polynomial by applyng the relation:

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

So you get:

`5x^2-17x+14=5(x-7/5)(x-2)=(5x-7)(x-2)`

Answer 2

You can use an AC method to find:

`5x^2-17x+14 = (5x-7)(x-2)`

Explanation:

This quadratic can be factored using an AC method:

Find a pair of factors of `AC = 5*15 = 70` with sum `B=17`

The pair `10, 7` works.

Use this pair to split the middle term and factor by grouping:

`5x^2-17x+14 = 5x^2-10x-7x+14`

`color(white)(5x^2-17x+14) = (5x^2-10x)-(7x-14)`

`color(white)(5x^2-17x+14) = 5x(x-2)-7(x-2)`

`color(white)(5x^2-17x+14) = (5x-7)(x-2)`