Try this beautiful problem based on Quadratic equation, useful for ISI B.Stat Entrance.
The equations \(x^2 + x + a = 0\) and \(x^2 + ax + 1 = 0\)
Algebra
Quadratic equation
Roots
Answer: (b)
TOMATO, Problem 240
Challenges and Thrills in Pre College Mathematics
Let the equations have a common root \(α\).Therefore \(α\) must satisfy two given equations.......
Therefore,
Now, \(α^2 + α + a = 0\)...................(1)
And, \(α^2 + aα + 1 = 0\).......................(2)
Can you find out the value of \(a\)?
Can you now finish the problem ..........
Therefore,
Using cross-multiplication betwwen (1) & (2) we will get.......
\(\frac{α^2}{(1 – a^2)} =\frac{ α}{(a – 1)} = \frac{1}{(a – 1)}\)
\(\Rightarrow {α}^2 = \frac{(1 – a^2)}{(a – 1) }=- (a + 1)\) & \(α=\frac{(a-1)}{(a-1)}=1\)
Now \({α}^2=(α)^2\)
\(\Rightarrow -(a+1)=1\)
\(\Rightarrow a = -2\)
Therefore (b) is the correct answer....
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