Question

Using complete sentences, explain what the discriminant is and what it tells you about the solutions of a quadratic equation. Provide a unique example to back up your explanation.

Answer 1

The discriminant is the value of b^2 - 4ac from the quadratic formula. It is the radicand in the formula.
If the discriminant is a perfect square and positive then you can take a square root that is a real number and your result is two rational numbers: Assume the equation x^2 - 3x + 2 = 0
then b^2 - 4ac = (-3)^2 - 4(1)(2) = 9 - 8 = 1 and there are two rational solutions
If the discriminant is 0, then the radical is 0. and you get a repeated rational solution
Assume that x^2 - 2x + 1 = 0, then b^2 - 4ac = (-2)^2 - 4(1)(1) = 0
the solution is x = -1 a repeated rational number
If the discriminant is positive but not a perfect square, you get two real but irrational solutions Assume x^2 - 4x + 2 = 0, b^2 - 4ac = (-4)^2 - 4(1)(2) = 16 - 8 = 8 and the solutions involves the +- sqrt (8)

If the discriminant is negative you get two complex solutions (since you have a negative under the square root). these solutions on conjugates of each other.
Assume x^2 - x + 1 = 0, b^2 - 4ac = (-1)^2 - 4(1)(1) = 1 - 4 = -3
the solutions are (1 +- sqrt(-3))/2 = (1/2) +- (sqrt 3)/2 i