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I don't really need the answer. I just need a quick review of how to factor.
I know the first one can't be factored.
is the second one supposed to be f(x)=-4x^2+16x-15 I put an x after the 16
I apologize, that was careless of me.
have you learned the quadratic formula?
I know the formula.
you can use the formula to solve -4x^2+16x-15 = 0 for x you'll get two roots x = p and x = q you can use those two solutions to find the factorization in the form k*(x-p)*(x-q) = 0 the k is some fixed number that scales the graph and determines whether the graph opens up or down
Oh my gosh. Thank you!
let me know what you get
x= 1.5 x=2.5
is this correct?
that's the same as x = 3/2 and x = 5/2
what we can do is multiply both sides of each equation by 2 x = 3/2 ----> 3x = 2 x = 5/2 ----> 5x = 2
then move the 2's over through subtraction 3x = 2 ----> 3x-2 = 0 5x = 2 ----> 5x-2 = 0
what would come next?
I'm not sure. I've never seen this done before.
oh wow I made a big typo I'm just noticing now
x = 3/2 should turn into 2x = 3
and x = 5/2 should turn into 2x = 5
Oh. Would you multiply them from there to get the original function?
ok so through the quadratic formula, we get these 2 solutions x = 3/2, x = 5/2 multiply both sides by 2 2x = 3, 2x = 5 then move everything to one side 2x-3=0, 2x-5=0
we have 2x-3=0, 2x-5=0 they would turn into (2x-3)*(2x-5) = 0 I think that's what you had in mind?
what would (2x-3)*(2x-5) expand out into?
that 16 should be negative try again
now compare 4x^2-16x+15 (what you just got when you expanded) with -4x^2+16x-15 (the original function). Are they the same? If not, what can we do to make them the same?
I don't think they are the same. I'm not sure. :[
Would you need a negative one in front of the one I got?
they aren't the same notice how the 4x^2 is positive in 4x^2-16x+15 then we have -4x^2 in -4x^2+16x-15
same for the 16x terms and the 15 terms too
Right. So...what did I do wrong?
Why aren't they the same?
well to easily fix this, we can stick a -1 in front of the factorization
since that will make +4x^2 turn into -4x^2 the -16x turn into +16x and the +15 turn into -15
so k = -1 is that constant I was talking about
Oh! I think I get it. I understand evrything except the constant. Why is that -1?
Where did the -1 come from?
if we expanded out (2x-3)*(2x-5) we end up with 4x^2 as one of the terms we want -4x^2 so we can make (2x-3)*(2x-5) negative and say -1*(2x-3)*(2x-5) instead to fix that issue
expanding out (2x-3)*(2x-5) will have -16x but we want +16x that -1 out front fixes the issue
Understood. Thank you so much, Jim!
You're the best! Have a good night!
you have a good night as well