SOLUTION: State whether the following statements are true or false and justify your answer. a.a(x-alpha)(x-beta) can always be expressed as a(x-h)^2+k b.a(x-h)^2+k can always be expresse

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: State whether the following statements are true or false and justify your answer. a.a(x-alpha)(x-beta) can always be expressed as a(x-h)^2+k b.a(x-h)^2+k can always be expresse      Log On


   

Question 193037: State whether the following statements are true or false and justify your answer.
a.a(x-alpha)(x-beta) can always be expressed as a(x-h)^2+k
b.a(x-h)^2+k can always be expressed as a(x-alpha)(x-beta).
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a) The given statement is true, here's why...


... Start with the given expression.


... FOIL


... Combine like terms.


Let and to get





... Take half of "b" and square it to get b%5E2%2F4. Add AND subtract this inside the parenthesis.


... Factor the first three terms in the parenthesis


... Distribute


Let to get





Let to get





So for ANY expression of the form you can rewrite it as



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b) The given statement is false (if you are only restricted to factor over the reals)


Here's a counter-example:


Let a=1, h=1, and k=3. So the general expression a%28x-h%29%5E2%2Bk becomes


%28x-1%29%5E2%2B3

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%28x-1%29%5E2%2B3 Start with the given expression.


x%5E2-2x%2B1%2B3 FOIL


x%5E2-2x%2B4 Combine like terms.


Since you CANNOT factor x%5E2-2x%2B4 over the reals, this means that %28x-1%29%5E2%2B3 CANNOT be written in the form of


Note: if you are allowed to factor over the complex numbers, then you can rewrite a%28x-h%29%5E2%2Bk into