Again, I have to ask a question... I fee like I'm giving you my all homework but... this website is my only way to understand and learn something! (I guess, people who answer my questions and explaine to me, do it better than my teacher...) thank you

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There are a couple of ways to "consider" factoring these.  They stem from:

y = ax^2 + bx + c

The general form for a quadratic equation, the form your equation is written in, with:

a = 1. b = -2, and c = -15

I will assume your teacher is working with the "easier method" first, when a = 1.


You are looking for two parenthesis of the form when you factor this:

(x + #)(x + #), both numbers can be different, even positive and negative.

When a = 1, then the method calls for trying to determine 2 numbers that multiply to c but add to b.  So, in this case:

two number that multiply to -15 and add to -2

Start considering all the factors for -15:
5*-3

-5*3

1*-15

-1*15

There are no other pairs of numbers.  The question, which pair of numbers, when you add them together, gives you -2?  The -5 and 3:

-5*3 = -15

-5 + 3 = -2

So, from there, you just plug them into (x+#)(x+#):

x^2 - 2x - 15 = (x + -5)(x + 3) = (x - 5)(x + 3)

So, we have:

y = (x - 5)(x + 3)

To find the horizontal intercepts, or the x intercepts, you are setting each parenthesis = 0 and solving for x:

x-5 = 0    and x + 3 = 0

So, x = 5 and -3

So, the graph would cross the x axis at (5,0) and (-3,0)

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The equation `h(z)=z^2-2z-15` is easily factored.  Two factors of 15 with a difference of 2 are: 5 and 3.  Since the second coefficient is negative, then the 5 has to be negative to get both coefficients to be negative.  The factored function is `h(z)=(z+3)(z-5)`

I infer from the second half of the question that they mean h(z)=y and z=x, and the horizontal intercepts are the x-intercepts i.e. where the graph of the function crosses the x-axis or y=0  There are exactly two coordinates where this happens, and in the ascending order of x values they are (-3,0) and (5,0). 

You can verify by plugging the first number of each ordered pair in for z and finding that the computed value for h(z) is 0 (the second number in each ordered pair). 

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