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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

- 1113
- 18
- 53361
- 5082

Equation

Integers

Roots

But try the problem first...

Answer: 18.

Source

Suggested Reading

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.

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