Condition for common root:
Consider ax2 + bx + c = 0 and dx2 + ex + f = 0 have a common root α. Then aα2 + bα + c = 0 and dα2 + eα + f = 0 computing for α2 and α, we have
=> (dc - af)2 = (bf - ce) (ae - bd)
which is the needed condition for the two equations to have a common root.
Important results:
- To search the common root of two equations, create the coefficient of second degree terms in both the equations same and then subtract the two equations. The value of x so calculated is the required general root.
- Condition for both the roots to be general is
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Illustration: If the relation of x2 + 2xsiny + 1 = 0, where y Î (0, p/2) and ax2 + x + 1 = 0 have a basic root, then calculate the value of a and y.
Key concept: If two quadratic relations with real coefficients have an imaginary root common, then both roots may be common and the two equations may be same. 
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Solution: Since discriminate of x2 + 2xsiny + 1 = 0 is 4sin2y - 4 < 0. Therefore roots of that equation are imaginary. Now that equation and ax2 + x + 1 = 0 have a general root, therefore roots of ax2 + x + 1 = 0 are also imaginary. That implies shown two equations have both the roots common. So both the equation is similar
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