Question

Factor 6x^2 -42x-54 Please show work

Answer 1

Notice that 6 is a factor of all three coefficients and thus can be factored out:

6x^2 -42x-54  =>  6(x^2 - 7x - 9).   x^2 - 7x - 9  does not factor so easily.  Remembering that obtaining roots of a quadratic makes it easy to write out the corresponding factors, let's apply the quadratic formula to x^2 - 7x - 9:

a=1, b = -7, c = -9. Then the discriminant is b^2 - 4ac, or (-7)^2 - 4(1)(-9), or 85.

Thus, the roots are:

      7 plus or minus √85        7 plus or minus 9.220

x = -------------------------------- = ------------------------------------

                 2                                           2

Let's call these roots "a" and "b."  Then the factors of the quadratic x^2 - 7x - 9 are (x-a) and (x-b), or, in this case, (x-8.11) and (x+1.11).

The original quadratic has three factors:  6, (x-8.11) and (x+1.11).