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Math Help - Need help. Sort of Urgent!

  1. #1
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    Need help. Sort of Urgent!

    This is a question on an AS core 1 paper. Im not really sure what its asking me to do.

    Find constants a and b, such that for all values of x:

    x^2 + 4x + 14 = (x + a)^2 +b
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  2. #2
    Super Member wingless's Avatar
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    x^2 + 4x + 14 = (x+a)^2 + b

    x^2 + 4x + 14 = \underbrace{x^2 + 4x + 4}_{\text{a square}} + 10 = (x+2)^2 + 10

    (x+2)^2 + 10 = (x+a)^2 + b

    And as you see here, we found a = 2 and b = 10
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  3. #3
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    Quote Originally Posted by nugiboy View Post
    This is a question on an AS core 1 paper. Im not really sure what its asking me to do.

    Find constants a and b, such that for all values of x:

    x^2 + 4x + 14 = (x + a)^2 +b
    Alternatively, a more general approach is to:

    1. Expand the right hand side and then
    2. Equate coefficients of powers of x on each side:

    Coef of x: 4 = 2a => a = 2.

    Coef of x^0 (ie the constant term): 14 = a^2 + b => 14 = 2^2 + b => 10 = b.
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  4. #4
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    Quote Originally Posted by nugiboy View Post
    This is a question on an AS core 1 paper. Im not really sure what its asking me to do.

    Find constants a and b, such that for all values of x:

    x^2 + 4x + 14 = (x + a)^2 +b


    There is a need to understand the statement correctly. I think Mr.F's suggestion should be followed because it is precise and complete.

    Idea:For two polynomials p and q of finite degree, if \forall x\in \mathbb{R} p(x) = q(x) holds, then they are identical polynomials.

    This idea is important because they are embedded within the definition of a polynomial. For polynomials to be well defined, we need a condition to know when they are same.

    So using this idea shows that you have understood the definition of polynomials.

    So write the statement:
    \forall x \in \mathbb{R} ,x^2 + 4x + 14 = (x + a)^2 +b = x^2 + (2a)x + (a^2 + b)
    This means the polynomials are identical and hence  2a = 4 and a^2+b = 14
    This method says this must be the ONLY solution.

    If you follow the eyeballing method, there is an additional question you have to ask yourself. and that is
    Introspection: "Oh alright, I can see one solution clearly, but what if there are other solutions? What if, for some other values of a and b, you get the same equation?".

    If you can see why there cant be other solutions, then this method is fine too. But can you?
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  5. #5
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    Isomorphism is right, however for the core 1 AS paper they like you to solve by inspection (weird I know but true). Indeed if the question proceeds to be more taxing then i would personally use the latter method.
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  6. #6
    Super Member wingless's Avatar
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    There can't be another solution because in order to get 4x, a must be 2. Then b has only one value too.

    I just thought that it was a 8th-9th grade question and he wouldn't know these things about polynomials. Surely mr fantastic's solution is more general and can be applied to other questions like this.
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  7. #7
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    Quote Originally Posted by wingless View Post
    There can't be another solution because in order to get 4x, a must be 2. Then b has only one value too.

    I just thought that it was a 8th-9th grade question and he wouldn't know these things about polynomials. Surely mr fantastic's solution is more general and can be applied to other questions like this.
    Yes ,of course, wingless.I have nothing against your solution.
    My opinions:
    I just thought I will educate the poster about the idea. He has already thanked you, so most likely he will not even read this. But in case some one else is reading this thread, I thought it will help them out. The different ideas for a solution and learning the pitfalls of junior class methods are important processes in learning.I wanted him to think, thats all.
    But in the hindsight, I think you are right, probably it is bad to over-instruct too

    Quote Originally Posted by Sean12345
    Isomorphism is right, however for the core 1 AS paper they like you to solve by inspection (weird I know but true). Indeed if the question proceeds to be more taxing then i would personally use the latter method.
    What is a "core 1 AS paper"? How much algebra do people at this level know?
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  8. #8
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    Quote Originally Posted by Isomorphism View Post

    What is a "core 1 AS paper"? How much algebra do people at this level know?

    It is the first exam british Students doing Advance Level math take it is required by all British universities for undergraduate degrees, but many require additional test.

    here are some sample papers
    AQA GCE A/AS Mathematics
    Edexcel : Qualifications : GCE including applied subjects
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  9. #9
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    Quote Originally Posted by bobak View Post
    In the first exam british Students doing Advance Level math take it is required by all British universities for undergraduate degrees, but many require additional test.

    here are some sample papers
    AQA GCE A/AS Mathematics
    Edexcel : Qualifications : GCE including applied subjects
    Then he must know what I am talking about. And if it is a subjective paper, he will be expected to write that!

    Thanks bobak for the information
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  10. #10
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    Quote Originally Posted by Isomorphism View Post
    Then he must know what I am talking about. And if it is a subjective paper, he will be expected to write that!

    Thanks bobak for the information
    The question posted really isn't a proper Advance Level it more of a confidence booster question so most people should be able to solve it by inspection.

    However the method i was taught for these problems was called "completing the square"

    which is pretty much this.
    x^2 + ax + b =  \left ( x + \frac{a}{2} \right )^2 - \left ( \frac{a}{2} \right )^2 +b

    nugiboy should be familiar with this method form doing GCSE maths.
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