i have very little information in my math book on the following questions. can you either tell me the answer or a link where i can find my own answer. thanks

Question

How do you know if a quadratic equation will have one, two, or no solutions?

How do you find a quadratic equation if you are only given the solution?

Answer 1

1. b^2 < 4ac: 2 imaginary solutions.

b^2 = 4ac: 1 real solution.

b^2 > 4ac: 2 real soutions.

2. Let's take a given Eq and solution
sets and use the solutions to derive the Eq.

Eq: y = 2x^2 + 5x + 2.

Solution set: x = -1/2, and x = -2.

x = -1/2,
x + 1/2 = 0.

x = -2,
x + 2 = 0.

(x + 1/2) (x + 2) = 0,
Multiply the 2 binomials:
x^2 + 2x + x/2 + 1 = 0,
x^2 + 5x/2 + 1 = 0,
Multiply both sides by 2 to eliminate the fraction and get:
2x^2 + 5x + 2 = 0.

The derived Eq is equivalent to the
given Eq.