Evaluate the discriminant of the equation. tell how many solutions each equation has and whether the solutions are real or imaginary. x^2 - 4x - 5 = 0

It has two distinct solutions and they are real.

Step-by-step explanation:

1. By definition: if the discriminant is negative, the quadratic equation does not have real solutions, it has two imaginary solutions. If the discriminant is zero the quadratic has one solution. If the discriminant is positive, the quadratic equation has two distinct solutions.

2. The discriminant is:

3. The discriminant is positive, therefore, the quadratic equation has two distinct solutions and they are real.

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240