Can You Factor Completely X^2 - 5x - 6 And Show All Steps?

X^2 - 5x - 6
x^2 - 6x + x -6
x(x-6)+1(x-6)
(x+1)(x-6)

Factor: X^2+5x-6

In the quadratic expression, you first look at the coefficient of x² and the constant (that is, the coefficient of x⁰). The signs of these will tell you what the signs of the factors need to be. In this case the coefficient of x² is positive (+1), so both factors will have (+1x) as one of the terms. The sign of the coefficient of x⁰ is negative, so one of the factors will be (x-something) and the other will be (x+something), like this: (x-a)(x+b).

If we multiply out what we just wrote, it will help us figure out what a and b need to be.
(x-a)(x+b) = x² + x(b-a) - ab.

This tells us that the two factors of -6 need to add to get -5. Let's look at the possibilities:
-6 = -1*6, sum of factors = +5
-6 = -2*3, sum of factors = 1
-6 = -3*2, sum of factors = -1
-6 = -6*1, sum of factors = -5. It looks like this might do it.

x² - 5x - 6 = (x - 6)(x + 1)
_____
Now that you have multiplied out (x - a)(x + b), you know the coefficient of x¹ in a quadratic is the sum of the factors of the coefficient of x⁰. Because you know this, you can "cut to the chase" when you do future problems.

If the coefficient of x² is not 1, then you have to work the factors of that coefficient into the mix. You can do the same sort of "multiply it out and see what you get" exercise we did above to give you a clue where to start.

Use diamonds and dust. It looks like an ╳, so the top portion would be -6 while vertically from the top would be -5. So the two spaces on the sides when multiplied together equals -6 but when added equals -5.
So the factor would be"
(x-1)(x+1)
and if you want to find the zeroes set the factors to 0.
X-1=0  x+1=0

add 1, subract one.
So the zeroes would be x= -1,1

(x+2)(x+3)