If a is an integer, for how many integral values of n can the quadratic equation x2 – (2a + 3)x + 4n = 0, have real and equal roots for x?

Question

If a is an integer, for how many integral values of n can the quadratic equation x2 – (2a + 3)x + 4n = 0, have real and equal roots for x?
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Solution 1

The quadratic equation is given as x^2 - (2a + 3)x + 4n = 0.

For a quadratic equation ax^2 + bx + c = 0 to have real and equal roots, the discriminant (b^2 - 4ac) must be equal to zero.

In this case, a = 1, b = -(2a + 3), and c = 4n.

So, we have [-(2a + 3)]^2 - 414n = 0.

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