Calculate the discriminant of a quadratic equation and determine the nature of its roots.
Last updated: March 2026 | By ForgeCalc Engineering
In algebra, the discriminant of a quadratic equation ax² + bx + c = 0 is the value b² - 4ac. It "discriminates" or distinguishes between the different types of roots the equation can have.
By looking at the sign of the discriminant, you can immediately tell if the quadratic has two real roots, one real root, or two complex roots, without actually solving the equation.
Where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
Find the discriminant of x² + 5x + 6 = 0:
1. Identify: a = 1, b = 5, c = 6
2. Calculate: D = 5² - 4(1)(6)
3. Simplify: D = 25 - 24 = 1
Final Answer: D = 1 (Two distinct real roots)
If D > 0, the quadratic equation has two distinct real roots. The graph of the parabola crosses the x-axis at two points.
If D = 0, the quadratic equation has exactly one real root (a repeated root). The vertex of the parabola touches the x-axis at exactly one point.
If D < 0, the quadratic equation has no real roots. Instead, it has two complex (imaginary) roots. The parabola never touches the x-axis.
The discriminant is the part of the quadratic formula that is under the square root symbol: $sqrt{b^2 - 4ac}$.
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