Please help me with this algebra question?!?!?!?

Find the discriminant of the equation in terms of p and q. Suppose q<p<0. Tell how many real solutions the equation has. px^2+qx-p

I really need help with this math question!

And indoor gym room consists of a rectangular region designated for excersice equipment and a semicircle on each end of the rrectangular region. The perimeter of the room is to be a 200 meter single-lane running track.

A. What is the radius of the semicircular ends of the room? Determine the distance, in terms of y, around the inside edge of the two semicircular parts of the track.
b. Use the result of part a to write an equation, in terms of x and y, for the distance traveled in one lap around the inside edge of the track. SOlve for y.
c. Use the result of part b to write the area A of the rectangular region as a function of x.
d. Rewrite the equation in part c in intercept form.
e. What dimensions will produce a maximum area of a rectangle?

Thank you so much!

I'd be happy to help you with your algebra and math questions!

For the first question, we have the equation px^2 + qx - p. To find the discriminant, we can use the formula b^2 - 4ac. In this case, a = p, b = q, and c = -p. Therefore, the discriminant is q^2 - 4p(-p) = q^2 + 4p^2.

To determine the number of real solutions, we can look at the discriminant. If the discriminant is positive, then there are two real solutions. If the discriminant is zero, then there is one real solution (also known as a repeated root). If the discriminant is negative, then there are no real solutions.

Now, let's move on to the second question about the indoor gym room.

a. To find the radius of the semicircular ends of the room, we need to find the perimeter of the room and equate it to the given value of 200 meters. Let's denote the width of the rectangular region as x. The perimeter consists of three parts: the length of the rectangular region (2x), the circumference of one semicircle (πr), and twice the radius (2r) due to having two semicircular ends. So we have the equation:

2x + πr + 2r = 200.

b. To find the distance around the inside edge of the two semicircular parts of the track, we need to sum the semicircular lengths. One semicircular length is πr, so the total distance would be 2πr.

c. To write the area A of the rectangular region as a function of x, we can multiply the length and width of the rectangular region: A = x * y.

d. To rewrite the equation from part c in intercept form, we need to solve for x and y in terms of A.

e. To find the dimensions that will produce the maximum area of a rectangle, we can use calculus and find the critical points of the function A. We can differentiate A with respect to x and set it equal to zero to find the critical points.

I hope this helps! Let me know if you have any further questions.