The Discriminant
The discriminant is a Grade 7 math skill from Yoshiwara Intermediate Algebra that determines the number and type of solutions to a quadratic equation ax^2 + bx + c = 0 using b^2 - 4ac. A positive discriminant means two real solutions, zero means one repeated solution, and negative means no real solutions.
Key Concepts
Property The discriminant of a quadratic equation is $$D = b^2 4ac$$ 1. If $D 0$, there are two unequal real solutions. 2. If $D = 0$, there is one solution of multiplicity two. 3. If $D < 0$, there are two complex conjugate solutions.
Examples For $y = x^2 x 3$, the discriminant is $D = ( 1)^2 4(1)( 3) = 13$. Since $D 0$, the equation has two distinct real solutions and the graph has two x intercepts.
For $y = 2x^2 + x + 1$, the discriminant is $D = 1^2 4(2)(1) = 7$. Since $D < 0$, the equation has two complex solutions and the graph has no x intercepts.
Common Questions
What is the discriminant?
The discriminant of ax^2 + bx + c = 0 is D = b^2 - 4ac. It tells you how many real solutions the quadratic has without fully solving.
What does each discriminant value mean?
D > 0: two distinct real solutions. D = 0: one repeated real solution. D < 0: no real solutions (complex).
How do you calculate the discriminant of x^2 - 5x + 6?
a=1, b=-5, c=6. D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0, so two real solutions.
Can the discriminant be used without the quadratic formula?
Yes. The discriminant alone tells you the number of solutions. You only need the full quadratic formula to find the actual solution values.