Question

How do you factor `24x^2-76x+40`?

Answer 1

First, let's simplify this dividing the equation by `4`.

`6x^2-19x+10`

Then, we have to find its roots: `x_1=2/3` and `x_2=5/2`. But in order not to work with these fractions, we can multiply both sides by the denominator of the fractions, leaving only rational numbers:

`3x_1=2/cancel(3)*cancel(3)` which means `3x_1-2=0`
`2x_1=5/cancel(2)*cancel(2)` which means `2x_2-5=0`

Now, let's just factor these.

`(3x-2)(2x-5)`

Done.

Answer 2

`24x^2-76x+40 = 4*6x^2-4*19x+4*10`

`= 4*(6x^2-19x+10)`

From the rational root theorem, if `p/q` is a root of `6x^2-19x+10 = 0` written in lowest terms, then `p` must be a divisor of the constant term `10` and `q` must be a divisor of the coefficient (`6`) of the highest order term.

Notice that `6` can be factored as `2*3` and `10` can be factored as `2*5`.

Let us try `(2x-5)(3x-2)`, it seems to work...

`(2x-5)(3x-2) = 2x*3x-5*3x-2x*2-5*(-2)`

`= 6x^2 - (15+4)x + 10`

`= 6x^2 - 19x + 10`

So in summary, `24x^2-76x+40 = 4*(2x-5)(3x-1)`

Answer 3

`24x^2-76x+40 = 4(6x^2-19x+10)`.

To factor `6x^2-19x+10` multiply `6xx10 = 60`.

We need two numbers that multiply to give us `60` and add to get us `-19` (the number in the middle).

A little thought should convince us that both numbers must be negative. We proceed in order:

`-1xx-60` do not add up to `-19`

`-2xx-30` do not add up to `-19`

`-3xx-10` do not add up to `-19`

`-4xx-15` STOP! These do add up to `-19`.

We'll split the `-19x` into `-4x-15x` (or `-15x-4x` either will work)

`6x^2-19x+10 = 6x^2-4x-15x+10`.

Now factor by grouping:

`6x^2-19x+10 = (6x^2-4x)+(-15x+10)`.

`6x^2-19x+10 = 2x(3x-3)+(-5)(3x-2)` .

`6x^2-19x+10 = (2x-5)(3x-2)`.

So we finish with:

`24x^2-76x+40 = 4(6x^2-19x+10)`

`color(white)"ssssssssssssssss"` `= 4(2x-5)(3x-2)`