Question

If \[a{x^2} + bx + c\] and \[b{x^2} + ax + c\] have a common factor \[x + 1\], hence prove that c = 0 and a = b.

Answer 1

Hint: Here, put x = − 1, in both the given equations to get the two linear expressions containing a, b and c only. Add the two equations obtained to get the value of c. then put the value of c in any of the equations containing a, b and c, you will get the relation between a and b.

Complete step-by-step answer:
We have two equations \[a{x^2} + bx + c\] and \[b{x^2} + ax + c\].
It is given that both equations have a common factor (x + 1) i.e., − 1 is root or zero of both the given equations.
As (− 1) is a root of \[a{x^2} + bx + c\]
\[a{( - 1)^2} + b( - 1) + c = 0\]
$\Rightarrow$ a – b + c = 0 …(i)
Also, (− 1) is a root of \[b{x^2} + ax + c\]
\[b{( - 1)^2} + a( - 1) + c = 0\]
$\Rightarrow$ b – a + c = 0 …(ii)
Adding equations (i) and (ii), we get
(a – b + c) + (b – a + c) = 0 \Rightarrow 2c = 0 \Rightarrow c = 0
Putting value of c in equation (i),
a – b = 0 \Rightarrow a = b
a = b also satisfies equation (ii), so for both equations a = b.
Therefore, if \[a{x^2} + bx + c\] and \[b{x^2} + ax + c\] have a common factor \[x + 1\], then c = 0 and a = b.

Note: In these types of questions, also form linear equations if any factor or zeroes are given. In this question the concept used is, if (x + 1) is a factor of the equation then – 1 is a zero of that equation or if we replace variables like x, y by – 1, we will get the value of expression as 0.