Question

Describe the discriminant of the related equation of the graph. Then determine the type and number of roots.

𝑏2−4⁢𝑎⁢𝑐 < > or = 0;

number of roots:

type of root(s):

Answer 1

For a quadratic ax^2 + bx + c the discriminant is
D = b^2 − 4ac.

Cases:

- D > 0
- Number of real x‑roots: 2 (two distinct real roots).
- Type: two distinct real numbers (if D is a perfect square and coefficients are rational/integers the roots are rational; otherwise they are irrational).
- Graph: parabola crosses the x‑axis in two points.

- D = 0
- Number of real x‑roots: 1 (one real root of multiplicity 2).
- Type: a repeated (double) real root x = −b/(2a).
- Graph: parabola is tangent to the x‑axis at its vertex.

- D < 0
- Number of real x‑roots: 0 (no real roots).
- Type: two complex conjugate nonreal roots.
- Graph: parabola does not meet the x‑axis.

(If a = 0 the equation is not quadratic and the discriminant formula does not apply.)