The document discusses the quadratic formula and the discriminant. The discriminant determines the nature of the roots of a quadratic equation. A positive discriminant that is a perfect square yields two real roots, a positive non-perfect square discriminant yields two real irrational roots, a zero discriminant yields one real rational root (a double root), and a negative discriminant yields two complex (imaginary) roots. Several examples of applying the discriminant to determine the root types are provided.

1. Quadratic Formula
& The Discriminant
Algebra 2

2. What is the discriminant?
𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎

3. Why do we want to know the value of the
discriminant?
 Positive, perfect square–
 2 Real, rational roots
 Positive, non-perfect square–
 2 Real, irrational
 Zero –
 1 Real, Rational (double root)
 Negative –
 2 Complex (imaginary) roots

4. Examples:
9𝑥2
− 3𝑥 − 8 = −10 −2𝑥2
− 8𝑥 − 14 = −6

5. Examples:
2𝑥2
+ 5𝑥 − 4 = 0 −7𝑥2
+ 16𝑥 = 8𝑥

6. Examples:
−6𝑥2
+ 7𝑥 + 3 = 0 2𝑥2
− 10𝑥 − 5 = 0

7. Examples:
Find the value of a such that 𝑎𝑥2
+ 3𝑥 + 5 = 0 has two
real roots (solutions).

8. Examples:
Find the value of a such that 𝑎𝑥2
+ 48𝑥 + 64 = 0 has one
real root (solution).

9. Examples:
Find the value of a such that 𝑎𝑥2
+ 3𝑥 − 6 = 0 has two
imaginary roots (solutions).

10. Examples:
Find the value of c such that 2𝑥2
− 6𝑥 + 𝑐 = 0 has two
imaginary roots (solutions).

11. Examples:
Find the value of c such that −4𝑥2
+ 8𝑥 + 𝑐 = 0 has two
real roots (solutions).