Question

Valuate the discriminant for the following equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers.

2x=−2x+1

a) Discriminant: 0, One distinct rational solution

b) Discriminant: 4, Two distinct rational solutions

c) Discriminant: 1, Two distinct irrational solutions

d) Discriminant: -4, Two distinct nonreal complex solutions

Answer 1

The discriminant of a quadratic equation determines the nature and number of its solutions. If positive, there are two distinct real solutions; if zero, there is one real solution; and if negative, there are two complex solutions.

Step-by-step explanation:

The discriminant of a quadratic equation ax²+bx+c = 0 is given by the expression b² - 4ac. Evaluating the discriminant can tell us about the nature and number of solutions to the quadratic equation. There are three possible cases:

• If the discriminant is positive, we have two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution, and the solution is also rational if the coefficients of the equation are rational numbers.
• If the discriminant is negative, there are two distinct nonreal complex solutions.

To determine if the real solutions are rational or irrational, we must examine the square root of the discriminant. If the square root is rational, then the solutions are rational. Otherwise, if the square root is an irrational number, the solutions are irrational.