Quadratic Discriminant & Types of Roots

Flashcards reviewing the discriminant, its conditions, and example applications for determining the nature of quadratic roots.

What is the formula for the discriminant of a quadratic equation ax² + bx + c = 0?

Δ = b² − 4ac

If the discriminant is greater than zero (Δ > 0), what is the nature of the roots?

Two real and distinct (different) roots

If the discriminant equals zero (Δ = 0), what is the nature of the roots?

One repeated real root (also called equal or double root)

If the discriminant is less than zero (Δ < 0), what is the nature of the roots?

Two complex (non-real) conjugate roots

What single inequality guarantees that a quadratic has real roots (either distinct or equal)?

Δ ≥ 0

What is meant by a "repeated" or "equal" root?

Both solutions are the same real number because Δ = 0

List the three main steps to determine the nature of the roots of a quadratic.

1) Rewrite the equation in standard form ax² + bx + c = 0; 2) Identify a, b, c; 3) Compute Δ = b² − 4ac and compare with >0, =0, or <0

How do you find the parameter values that give equal roots for a quadratic with a parameter?

Set the discriminant equal to zero (Δ = 0) and solve for the parameter

For x² + 2mx + (8m − 15) = 0 to have equal roots, what equation in m must be satisfied?

4m² − 32m + 60 = 0 (or simplified m² − 8m + 15 = 0)

What values of m make x² + 2mx + (8m − 15) = 0 have equal roots?

m = 3 or m = 5

For (k + 5)x² − 8x + 1 = 0 to have real roots, what inequality must k satisfy?

k ≤ 11

For x² + kx + 3 = 0 to have one repeated root, what are the possible k values?

k = ±2√3

Why must a quadratic be written with the right-hand side equal to zero before applying the discriminant test?

Because the coefficients a, b, and c are defined from the standard form ax² + bx + c = 0

Which discriminant condition corresponds to "real and distinct" roots?

Δ > 0