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.Com
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) (Show Source): You can put this solution on YOUR website!
a) The given statement is true, here's why...
(x-\beta))
... Start with the given expression.
)
... FOIL
 x +\alpha\beta))
... Combine like terms.
Let )
and 
to get
)
)
... Take half of "b" and square it to get . Add AND subtract this inside the parenthesis.
^2-\frac{b^2}{4}+c\right))
... Factor the first three terms in the parenthesis
^2+a\left(-\frac{b^2}{4}+c\right))
... Distribute
Let )
to get
^2+k)
Let 
to get
^2+k)
So for ANY expression of the form you can rewrite it as
============================================================
b) The given statement is false (if you are only restricted to factor over the reals)
Here's a counter-example:
Let , , and . So the general expression becomes
-----------
Start with the given expression.
FOIL
Combine like terms.
Since you CANNOT factor over the reals, this means that CANNOT be written in the form of
Note: if you are allowed to factor over the complex numbers, then you can rewrite into
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