Question

If the roots of a quadratic equation are m + n and m - n, then the quadratic equation will be:
1. \(\rm x^2 +2\ mx + m^2 - mn+n^2=0\)
2. \(\rm x^2 +2 \ mx+(m-n)^2=0\)
3. \(\rm x^2-2 \ mx+m^2-n^2=0\)
4. \(\rm x^2+2 \ mx+m^2-n^2=0\)
5. None of these

Answer 1

Correct Answer - Option 3 : \(\rm x^2-2 \ mx+m^2-n^2=0\)

Concept:

Factor theorem:

If \(\alpha\) is a root of the polynomial p(x) then \(\rm (x-\alpha)\) is a factor of p(x).

Calculation:

Since m + n and m - n are roots of the quadratic equation, by factor theorem, (x-(m-n)) and (x-(m+n)) are the factors of the quadratic equation.

Therefore, the quadratic equation is given by:

\((x-(m-n))(x-(m+n)) = x^2-x(m+n)-x(m-n)+(m+n)(m-n)\)

\(= x^2 - x(m+n +m-n)+(m^2 - n^2) \)

\(\\ = x^2 - 2mx+(m^2-n^2)\)

Therefore, the required equation is \(\rm x^2-2mx+(m^2-n^2)\).

Note that as all the options have leading coefficient 1 we can directly calculate the equation by simply multiplying the factors.