1Cademy - Number of Solutions of a Quadratic Equation Based on the Discriminant

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Number of Solutions of a Quadratic Equation Based on the Discriminant

The sign of the discriminant $b^2 - 4ac$ reveals how many real solutions a quadratic equation $ax^2 + bx + c = 0$ (with $a \neq 0$ ) has, without requiring the equation to be fully solved:

If $b^2 - 4ac > 0$ (positive discriminant), the equation has two real solutions. A positive value under the square root yields two distinct results from the $\pm$ in the formula.
If $b^2 - 4ac = 0$ (zero discriminant), the equation has one real solution. The square root of zero is zero, so the $\pm$ produces only one value.
If $b^2 - 4ac < 0$ (negative discriminant), the equation has no real solutions. The square root of a negative number is not real, so neither branch of the $\pm$ yields a real result.

For example, $2x^2 + 9x - 5 = 0$ has discriminant $9^2 - 4 \cdot 2 \cdot (-5) = 121 > 0$ , so it has two solutions. The equation $4x^2 - 20x + 25 = 0$ has discriminant $(-20)^2 - 4 \cdot 4 \cdot 25 = 0$ , so it has one solution. And $3p^2 + 2p + 9 = 0$ has discriminant $2^2 - 4 \cdot 3 \cdot 9 = -104 < 0$ , so it has no real solutions.

Updated 2026-04-21

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