12. The g.c.d of the equations 2x2-x-1= 0 and 4x2+8x+3 = 0 is 3x+1
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Answer:Zero.
Let's assume both equations have a common root = r.
Since r is a root for both equations, it would individually satisfy both equations. So,
r^3 + 4r^2 + 5r + 18 = 0 --- (eq 1)
r^3 + 3r^2 + 10r + 12 = 0 --- (eq 2)
Clearly, eq1 = eq2 = 0
So,
r^3 + 4r^2 + 5r + 18 = r^3 + 3r^2 + 10r + 12
which leads to a quadratic equation
r^2 -5r + 6 = 0
Its solutions are 2, 3
But you will very easily see conflict, none of these solution (2 or 3) satisfies eq1 or eq2. So, our very first assumption of having a common root was wrong.
Step-by-step explanation:
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