Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Equations and Roots.
Consider the equation of the form \(x^{2}+bx+c=0\). The number of such equations that have real roots and have coefficients b and c in the set {1,2,3,4,5,6}, (b may be equal to c), is
Equation
Integers
Roots
But try the problem first...
Answer: 18.
B.Stat Objective Problem 123
Challenges and Thrills of Pre-College Mathematics by University Press
First hint
We know that if a quadratic equations have real roots then it's discriminant is >=0 so here \( b^{2}-4c \geq 0\). Now we will go casewise . First we will choose a particular Value for b then check what are the values of c that satisfies the above inequality.
Second Hint
for b=2, c=1
for b=3, c=1,2
for b=4, c=1,2,3,4
for b=5, c=1,2,3,4,5
for b=6, c=1,2,3,4,5,6
Final Step
we get required number =18.
I.S.I. & C.M.I. Entrance Program
Taught by students and alumni of I.S.I. & C.M.I.
