Practice Assignment 4 Question 9

[CG]

Question : Suppose one root of a quadratic equation of the form ax2+ bx + c = 0, with a , b , c ∈ R, is 2 + √3 . Then choose the correct set of options

1.  There can be infinitely many such quadratic equations.
2.  There is no such quadratic equation
3.  There is a unique quadratic equation satisfying the properties.
4.  x2 − 4x + 1 = 0 is one such quadratic equation
5.  x2 − 2x − 3 = 0 is one such quadratic equation

Option 1 and 4 are the marked correct answers.

By substituting 2 + √3 as x in x2 − 4x + 1 = 0 it is possible to check validity of option 4. 

I am a bit confused on the proof as to why there can be infinitely many such quadratic equations. ie: Why is option 1 correct? What are the other quadratics that may be of the form ax2+ bx + c = 0 AND have 2 + √3 as one root?

[Debajyoti Biswas]

One root of the quadratic equation is known.

The other root can be any real number k.

For each value of k, we will have a different quadratic equation. therefore, there can be

infinitely many quadratic equations.