Try this beautiful problem Based on Quadratic equation, useful for ISI B.Stat Entrance.

## Quadratic equation | ISI B.Stat Entrance | Problem 198

Consider the quadratic equation of the form \(x^2 + bx + c = 0\). The number of such equations that have real roots and coefficients b and c from the set \(\{1, 2, 3, 4, 5\}\) (b and c may be equal) is

- (a) \(18\)
- (b) \(15)\)
- (c) \(12\)
- (d) None of these

**Key Concepts**

Quadratic equation

Algebra

Roots of the nature

## Check the Answer

But try the problem first…

Answer: (c) \(12 \)

TOMATO, Problem 198

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

The given equation is \(x^2 + bx + c = 0\).

we know that in the equation \(ax^2+bx+c=0\) ,the condition for real root is \(b^2-4ac \geq 0\)

Can you now finish the problem ……….

Final Step

Now, \(b^2 > 4c\)

b cannot be equal to 1.

If b = 2, c = 1

If b = 3, c = 1, 2

If b = 4, c = 1, 2, 3, 4

If b = 5, c = 1, 2, 3, 4, 5

Total number of equations = \(1 + 2 + 4 + 5 = 12\)

## Other useful links

- https://www.cheenta.com/problem-based-on-triangle-prmo-2018-problem-13/
- https://www.youtube.com/watch?v=pLAMlNUOdTs

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