why is the answer (B)?

[Sarah.N]

the answer of this question is B, but I don't know why.
if alpha and beta are 0, then x+1 should be 0.
so x would be -1.
but according to the question, for every x, the equation should be true.

 

[pka]

the answer of this question is B, but I don't know why.
if alpha and beta are 0, then x+1 should be 0.
so x would be -1.
but according to the question, for every x, the equation should be true.
View attachment 28364

You please give us at least one set of values of [imath](\alpha,\beta)[/imath] such that
[imath]({\large\bf\forall}| x\in\Re)[\alpha(x+1)^2+\beta(x+1)+x+1=0][/imath]
The we can discuss the matter further.

 

[topsquark]

I take it you meant

You please give us at least one set of values of [imath](\alpha,\beta)[/imath] such that
[imath]({\large\bf\forall}| x\in\Re)[\alpha(x+1)^2+\beta( x^2-3x - 2 )+x+1=0][/imath]
The we can discuss the matter further.

@Sarah.N:

Hint:
Can you solve your equation for [imath]\beta[/imath] in terms of [imath]\alpha[/imath] and x? If there are no solutions the x's will not cancel out. If there is one or more solutions all of the x's will cancel out. Do you see why?

-Dan

 

[Dr.Peterson]

the answer of this question is B, but I don't know why.
if alpha and beta are 0, then x+1 should be 0.
so x would be -1.
but according to the question, for every x, the equation should be true.

View attachment 28364

I think you missed a key part of the condition: The equation must be true for all x in R, not just for some x (-1 in your example). So your example does not contradict the claim that there are no values of alpha and beta for which the condition is true.

I hope that clarification of the problem will help you see what you need to do to solve it.

 

[Sarah.N]

I think you missed a key part of the condition: The equation must be true for all x in R, not just for some x (-1 in your example). So your example does not contradict the claim that there are no values of alpha and beta for which the condition is true.

I hope that clarification of the problem will help you see what you need to do to solve it.

I know this and that is exactly why I said B cannot be the correct answer. Because if B is the correct answer, then x should be only -1.
but the question said that the equation must be true for all x in R, so according to this, how can B be the correct answer?

I take it you meant


@Sarah.N:

Hint:
Can you solve your equation for [imath]\beta[/imath] in terms of [imath]\alpha[/imath] and x? If there are no solutions the x's will not cancel out. If there is one or more solutions all of the x's will cancel out. Do you see why?

-Dan

I think i get it but not sure
Does the question mean how many values there are for alpha and beta? I thought B says beta and alpha are zero

 

[Dr.Peterson]

Does the question mean how many values there are for alpha and beta? I thought B says beta and alpha are zero

Yes, it is asking for the number of values, not what those values are. B doesn't say that alpha and beta are zero; it says there are no values of alpha and beta that work.