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AMC-8 Math Olympiad

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

Try this beautiful problem from TOMATO useful for ISI B.Stat Entrance based on Quadratic Equation. You may use sequential hints to solve the problem.

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


But try the problem first…

Answer: (b)

Source
Suggested Reading

TOMATO, Problem 240

Challenges and Thrills in Pre College Mathematics

Try with Hints


First hint

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 ……….

Final Step

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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