If the quadratic equations, ax^2 + bx + c = 0 and bx^2 + cx + a = 0, where a, b, c are distinct, have one common root, then what is the common root?
Answer (2)
The common root of the quadratic equations a x 2 + b x + c = 0 and b x 2 + c x + a = 0 is r = 1 . This is derived by setting both equations equal to their respective definitions and finding that both yield the same result when substituting r = 1 . Hence, the answer is confirmed by substituting back into the original equations.
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To find the common root between the two quadratic equations a x 2 + b x + c = 0 and b x 2 + c x + a = 0 , we can solve them under the condition that there is exactly one common root.
Let's denote the common root by α . Since α is a root of both equations, it should satisfy:
a α 2 + b α + c = 0
b α 2 + c α + a = 0
Now, subtract the first equation from the second equation:
( b α 2 + c α + a ) − ( a α 2 + b α + c ) = 0
This simplifies to:
( b − a ) α 2 + ( c − b ) α + ( a − c ) = 0
Since a , b , and c are distinct, the coefficients of this quadratic equation are not all zero. For these equations to have exactly one common root, one way is by using the condition that the discriminant of a quadratic equation equals zero, leading to a repeated root:
Therefore, one common and repeated root can be α = 1 .
Plug α = 1 into any of the equations, say the first one:
a ( 1 ) 2 + b ( 1 ) + c = 0
a + b + c = 0
Hence, when a + b + c = 0 , the common root is α = 1 . Therefore, this condition guarantees that 1 is a common root of both quadratics if they share only one common root.
