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

Quadratic equation

Roots

## Check the Answer

Answer: (b)

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….

## Other useful links

- https://www.cheenta.com/length-of-a-tangent-amc-10a-2004-problem-22/
- https://www.youtube.com/watch?v=pYSIvF7jZy4