Discriminant
Determines the nature of the roots of a quadratic equation.
Formula
Live Calculator
MembersVariables
Example
Δ > 0: two real roots; Δ = 0: one root; Δ < 0: complex roots
How it works
For a quadratic equation ax^2 + bx + c = 0, the discriminant is the part under the square root in the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Because square roots of positive, zero, and negative numbers behave differently, the value of b² - 4ac tells you whether the quadratic has two distinct real solutions, one repeated real solution, or no real real solutions. It works by measuring how far the parabola sits from the x-axis at its closest point.
When to use it
Use the discriminant whenever you want to know how many solutions a quadratic equation has without fully solving it. It is especially useful in algebra, graphing parabolas, and checking whether a word problem or model produces real-valued intersections.
Worked example, step by step
Find the discriminant of 2x^2 - 3x + 5 = 0 Identify coefficients: a = 2, b = -3, c = 5 Substitute into Δ = b^2 - 4ac Δ = (-3)^2 - 4(2)(5) Δ = 9 - 40 Δ = -31 Since Δ < 0, the equation has no real roots; its solutions are complex.
Common mistakes to avoid
- Forgetting to use the full b value, including its negative sign
- Mixing up the discriminant with the quadratic formula denominator 2a
- Assuming a negative discriminant means the equation has no solutions at all instead of no real solutions
Did You Know?
The single sign of the discriminant tells you the whole story: positive means two real roots, zero means one, negative means the roots are imaginary.
Share this formula
More in Algebra
View allQuadratic Formula
IntermediateFinds the roots of any quadratic equation ax² + bx + c = 0.
Binomial Theorem
IntermediateExpands any power of a binomial into a sum of terms.
Logarithm Product Rule
IntermediateThe logarithm of a product equals the sum of logarithms.
Logarithm Quotient Rule
IntermediateThe logarithm of a quotient equals the difference of logarithms.
Logarithm Power Rule
IntermediateThe logarithm of a power equals the exponent times the logarithm.
Change of Base Formula
IntermediateConverts a logarithm from one base to another.