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Quadratic equation | ISI-B.stat | Objective Problem 240

Try this beautiful problem based on Quadratic equation, useful for ISI B.Stat Entrance.

Quadratic equation | ISI B.Stat Entrance | Problem 240

The equations $x^2 + x + a = 0$ and $x^2 + ax + 1 = 0$

• (a) cannot have a common real root for any value of a
• (b) have a common real root for exactly one value of a
• (c) have a common root for exactly two values of a
• (d) have a common root for exactly three values of a.

Key Concepts

Algebra

Roots

TOMATO, Problem 240

Challenges and Thrills in Pre College Mathematics

Try with Hints

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….