New and clever SAT problem

[kweaver]

For your solving minds...

[BATcher]

Nobody seems to have bitten yet, so I say

the value of b can be anything you like - plug x=0 into the expression, and the value of b does not matter...

[kweaver]

Well, (a) that's not an answer option and (b) it's not the case. There is a specific value for b.

[mngerhold]

I am puzzled: I can work out that setting b to one of those values makes the expression simple (and I know which one), but cannot see why it has to be so. It all boils down to the exact meaning of the phrase "is equivalent to bx". If that means simply numerically equivalent, then (surely?) the equation re-arranges to:

x^2(4a - 1) + x(4a-4-b) = 0

and the solution for x is that it equals 0 or -(4a-4-b)/(4a-1). So 'b' can have any value we like. But that is clearly not what the puzzler intends. If 'equivalent' means 'algebraically equivalent' (if that is a valid term), then
x^2(4a-1) + x(4a-4) == bx
and then

with no x^2 term on the RHS, we know a must be 1/4, so
x(4a-4) = x(-3) == bx, so b = -3.

Maybe I need to go and read some SAT test rules, but I don't like this! I may have to accept I am just not clever enough. I am certainly not clever (or informed) enough to create a spoiler box as BATcher did - oh, I just found out something - who knew that option was there? (I did a reply with quote to see what Batcher had used).

Martin

PS Glad to see a puzzle here after so long. I do have one of my own I rather like...

[wavy]

we know a must be 1/4

I am missing this
I used to be good at this sort of thing
The stein is BACK

(4x+4)(ax-1)-x^2+4=bx

[wavy]

http://www.wolframalpha.com/input/?i...t+does+b+equal

wolframalpha could not get it either

[kweaver]

This is, what I consider, an awful problem from the College Board. However, the "key" to solving this is realizing that "b" is a constant (one of the values in the answer choices) and is the coefficient of "x". Therefore, you have to clear everything on the left side of the equation EXCEPT for some number of "x" values.

That means:

a = 1/4, since 4ax^2 - x^2 has to "go away" or be equal to 0. To do that, a has to be 1/4

[BATcher]

I'm not sure that you have told us what is the value of b, which is the original question!

[kweaver]

-3

4ax^2 - 4x + 4ax - 4 - x^2 + 4 = bx
cancel +4 and -4
Then: 4ax^2 - x^2 = 0 which means a = 1/4
Then, -4x + 4(1/4)x = -3x

[mrjimphelps]

x has to be -1, in order for b to be -3. If these are true, then you end up with -4a + 1 = 1 - 4a.

[kweaver]

Since the problem stated that the expression equals "bx" you're then looking for the coefficient of the "x" term. When you expand the problem, the -4 and +4 cancel. That, I suspect, was the College Board's attempt at a clue. The x^2 terms must also be eliminated because the expression = "bx" and there are no x^2 terms.
So, if 4ax^2 - x^2 must = 0 then, 4ax^2 = x^2 and 4a = 1, so a = 1/4. Consequently, 4ax - 4x becomes 4(1/4)x - 4x = x - 4x = -3x; therefore, b = -3.

[mrjimphelps]

Since the problem stated that the expression equals "bx" you're then looking for the coefficient of the "x" term. When you expand the problem, the -4 and +4 cancel. That, I suspect, was the College Board's attempt at a clue. The x^2 terms must also be eliminated because the expression = "bx" and there are no x^2 terms.
So, if 4ax^2 - x^2 must = 0 then, 4ax^2 = x^2 and 4a = 1, so a = 1/4. Consequently, 4ax - 4x becomes 4(1/4)x - 4x = x - 4x = -3x; therefore, b = -3.

Are you sure you aren't making an unjustified assumption here about simply eliminating the x^2 terms?

[kweaver]

Don't think so. Since the expression was equal to "bx", you need to convert the expression to have only an "x" variable. That means eliminating the constants and the x^2 terms.

[mrjimphelps]

Of course, you can't just eliminate the constants and the x^2 terms; you have to eliminate them in a mathematically-valid way. I'm sure you know that; I just thought I would point it out.