Concept 4.4

The Discriminant and the Nature of Roots

Source: NCERT Mathematics, Class 1010 minute lesson

Formal statement

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. If D > 0 there are two distinct real roots; if D = 0, two equal real roots; if D < 0, no real roots.

The intuition

D is what sits under the square root in the quadratic formula. A positive number has two square roots, zero has one, and a negative number has none among the real numbers.

So you can know how many roots there are without finding them. Two quantities are competing: b² pushes the roots apart, 4ac pulls them together, and D is the score.

The attention

Look at the sign of D, not the size of any one part.

x² + 2x + 5: D = 2² − 4·1·5 = −16

Look hereNot here

Look here: The sign of D. Only the sign matters: −16 is negative, so there are no real roots. You never need the roots themselves.
Not here: b² on its own. b² is never negative, so on its own it makes every equation look as if it has roots. It is only half of the contest.
What it teaches: Sometimes a question only needs the sign of an answer. Look for the quickest thing that decides it.