<p>(x-8)(x-k) = x^2 - 5kx +m</p>
<p>In the equation above, K and M are constants. If the equation is true for all values of x, what is the value of m?</p>
<p>a. 8
b. 16
c. 24
d. 32
e. 40</p>
<p>I think you are missing sth.
If ,as the problem says,the equation si true for all values of x ,assume that x = 8
==>
(8-8)(8-k) =8 ^2 -5k8 +m
0 x (8-k) =64-40k+m
0=64-40k+m
-64=-40k+m
You cannot possibly know tha value of m,it depends on k.</p>
<p>No Ivan, the question is not missing anything</p>
<p>Model:
(a+b)(a+c) = a^2 +ac+ab + ac</p>
<p>In this question it is asking about "m."
If you foil out the equation they give you, you get x^2 -kx + -8x + 8k. So since m is that last term, m = 8k. </p>
<p>In the answer choices they give you possible values of m, if m were choice 1 (8) then k would have to equal 1 (because 8k = m.... 8k = 8). Similarly, if m were choice 2 (16), k would have to be two (because 8k = 16). </p>
<p>So now we have prospective values of k. We also know that the middle mesh (kx + -8x) must = 5kx, according to the problem they give.</p>
<p>If m were 8, then k would be 1. Plug 1 into our middle terms and you get: (1)x + -8x, which is -7x. Now you have to find the combination of m and k which yields 5x.</p>
<p>if m is 24 (choice c), then k must equal three. Plug that into our middle terms and you get (3)x + -8x, which equals 5x.</p>
<p>sounds reaseonable.It is hard to solve problems at 1 .00 a.m :D at least tried: }</p>
<p>it says choice b is the correct answer (16)</p>
<p>Whoops, sorry, I read it as 5x. Its (B), 16, because when m = 16 then k = 2. (8k = 16).</p>
<p>The middle term has to = 5kx, which after you plug in 2 for k, turns into 10x.
Now go into the middle terms of the foiled out equation and you put in 2 for k... -(2)(x) -8x, which yields 10x.</p>
<p>Yeah. Dagol, you were right up until your last part where you lost me (a bit too lazy to figure out exactly what you were doing). Here is how to solve the problem:</p>
<p>Like Dagol said, foil it. x^2 - kx - 8x + 8k = x^2 - 5kx + m
Thus, m = 8k and -kx - 8x = -5kx
-8x = -4kx
x's cancel out, k = 2</p>
<p>m = 8k
m = (8)(2) = 16</p>
<p>EDIT: Well, Dagol posted a correction. Either way, here's how to solve it without plugging in.</p>
<p>Lol, that is much simpler.</p>
<p>^Latency, I understand everything you did except for "Thus, m = 8k and -kx - 8x = -5kx
-8x = -4kx." Would you mind elaborating how you arrived at those equations? Also, what did you do with the m (how did you get rid of it in the -kx-8x=-5kx-8x part?) Thanks a lot!</p>
<p>In the problem we know, (x-8)(x-k) = x^2 - 5kx +m. If you foil the right side you get
x^2 -kx-8x+8k = x^2 - 5kx +m.. Since they are equal we get set each individual part equal to each other.
so: x^2 = x^2
(-kx-8x) = -5kx
8k = m. </p>
<p>So what Latency did was set -kx-8x = -5kx and solve. Add +kx to the right and you get
-8x = -4kx. Divide by -4x and you get k = 2. According to our other equation, 8k =m. k =2, thus m = 16.</p>
<p>Ok I understand how he did it now, thanks. I never really thought/honestly didn't know that you could set the different parts of the equation equal to each other like that. So just so that I have this concept right, I can do this, right? :</p>
<p>Say the equation is 2x-6y+13 = 22-7g+v
I could do 2x=22, -6y=-7, 13=v? The signs have to be the same right? like if it was 2x- and on the other side it was 22+, I couldnt set the -6y = 7g? thanks again</p>
<p>A little faster:
applying Vieta's Theorem to a quadratic equation,
8+k = 5k
8k = m
k=2, m=16.</p>
<p>@Sasquatch</p>
<p>I don't think that's the same thing. In an equation like that, there is an infinite amount of possible values for x, y, g, and v. If you set each term equal in that equation, I guess that would give you one possible answer out of an infinite amount. </p>
<p>Although I'm not completely sure, I think that the reason you can set different parts of the equation equal in the other example is because each part that is set equal to each other is at a different power. I.e. the x^2s are put together, the x's are put together, and the constants are put together.</p>
<p>I probably wouldn't try solving any problems with this method unless it looks like this and is just asking for this style of solving it.</p>
<p>EDIT: Ah yes, let me reaffirm my response as to why you can do what I did for this problem. The key is that it says K and M are constants (K and M must be a solid number. K and M cannot equal 3x or 8x or 34x^2 or anything). Therefore the power of each term HAS to be constant. Thus, x^2 HAS to equal x^2. -kx -8x HAS to equal -5kx and 8k HAS to equal m. See, this wouldn't be true if K and M weren't constants because you could get a whole bunch of wild answers by having K and M equal some variable number such as 2x or 3x.</p>
<p>Well put Latency, basically this is only really applicable for quadratic equations. I have never seen it in any other situation.</p>
<p>I think I've seen a question in one of my SAT books something like
if (x-1)(x-2)(x-3)=x^3+mx^2+nx+p, what is the value of p?</p>
<p>What's Vieta Theorem?</p>
<p>(x-8)(x-k) = x^2 - 5kx +m</p>
<p>If x=0 : (-8)(-k) = m = 8k</p>
<p>Using foil: x^2 - (8+k)x + 8k = x^2 - 5xk + m</p>
<p>Remove equal terms: 8x + kx = 5kx</p>
<p>Divide out x and solve for k: k = 2</p>
<p>Substitute k : m = 8(2) = 16</p>
<p>I think the problem here is that people are matching up different parts of the equation and assuming they're equal. That may work on a test like this, but I don't think it's mathematically justified.</p>
<p>Haha I gave my algebra II teacher this problem today to see how he would solve it, and it took him about 15 minutes before he finally arrived at the correct answer. Later, I gave it to my precalc teacher and he solved it in about 15 seconds. Really interesting how some people can see the connections right away and others have to think a lot before they get it. Definitely a good question though.</p>