A very simple question. How do we prove it? a x 2 + b x...

Alissa Hancock

Alissa Hancock

Answered

2022-07-07

A very simple question. How do we prove it?
a x 2 + b x + c = a ( x r 1 ) ( x r 2 )

Answer & Explanation

Nathen Austin

Nathen Austin

Expert

2022-07-08Added 14 answers

For quadratic equation
0 = a x 2 + b x + c = a ( x + b 2 a ) 2 b 2 4 a c 4 a 2 ,
its roots are
r 1 , 2 = b ± b 2 4 a c 2 a .
Depending on the sign of b 2 4 a c, the roots could be distinct reals, a multiple real, or distinct complexes. We actually do not need the fundamental theorem of algebra here, which is usually used for equations of higher orders (e.g., order 5 or more).
Then we can evaluate it directly
a ( x r 1 ) ( x r 2 ) = a [ x 2 ( r 1 + r 2 ) x + r 1 r 2 ] = a [ x 2 ( b a ) x + b 2 ( b 2 4 a c ) 4 a 2 ] = a ( x 2 + b a x + c a ) = a x 2 + b x + c .
I guess people are overthinking about this question.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?