Why isn't y=a(x^2-Sx+P) same with x^2-Sx+P

Alexander Lewis 2022-10-24 Answered
Why isn't y = a ( x 2 S x + P ) same with x 2 S x + P
If we have roots of the function y = a x 2 + b x + c we can calculate S = b a and also P = c a . Then we know that we can form the function this way:
x 2 S x + P
So on the other side we know that we have the function f ( x ) = y in different ways:
y = a x 2 + b x + c
( α and β are roots of the quadratic function)
y = a ( x α ) ( x β )
And my question is here:
y = a ( x 2 S x + P )
Actually know that how we can form the qudratic equation using x 2 S x + P, but the function must be like y = a ( x 2 S x + P ). Actually I don't know that why we add a. I know it will be removed when ( a ) ( b a ). But I don't know that what is y = a ( x 2 S x + P ) different whitout a!
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Answers (2)

Dana Simmons
Answered 2022-10-25 Author has 14 answers
Step 1
a is there just to include the possibility of a quadratic having a coefficient other than 1 for 2 n d degree term x 2 .
Otherwise, how will you make a quadratic of the form:
y = ( a x 2 + b x + c )
(where a 1)
merely by taking it like:
y = ( x α ) ( x β ) ?
Step 2
Clearly, coefficient of x 2 is 1 in this.
So, while assuming a quadratic when its roots are known, we take it
y = a ( x α ) ( x β ) ,
just to be on the safer side.
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Christopher Saunders
Answered 2022-10-26 Author has 6 answers
Step 1
Two quadratic equations
y 1 = ( x b ) ( x c )
and
y 2 = a ( x b ) ( x c )
have the same roots but not the same values at any other points. They are different function which share common roots.
For example
y 1 = x 2 + 5 x + 6
and
y 2 = 3 x 2 + 15 x + 18
are different functions with the same roots.
Step 2
Notice that
y 1 ( 2 ) = 20
while
y 2 ( 2 ) = 60
What you like to say is that the two quadratic equations,
x 2 + 5 x + 6 = 0
and
3 x 2 + 15 x + 18 = 0
are equivalent because we can factor 3 out and 3 is not zero so it does not change the roots.
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