Question

The death drop is the tallest water slide in the park. The height of a rider can be expressed by the equation (-3x^2 + 9x = 54), where x is the number of seconds the rider has been on the ride. Factor the equation by first factoring out the gcf. What are the roots of this equation?

A) x=−3,6
B) x=3,−6
C) x=−2,9
D) x=2,−9

Now factor the equation without factoring out the gcf. What are the roots now? Did they change?
A) Yes, the roots changed.
B) No, the roots remained the same.
C) Cannot be determined.
D) The roots are imaginary.

How much time will it take the rider to reach the bottom of the slide?
A) x=−2 seconds
B) x=2 seconds
C) x=3 seconds
D) x=−3 seconds

Answer 1

The roots of the equation are x = 6 and x = -3, both obtained by factoring out the GCF or by factoring directly. Factoring out the GCF does not change the roots. The positive root x = 6 seconds is the time it takes for the rider to reach the bottom of the slide.

Step-by-step explanation:

The height of a rider can be expressed by the quadratic equation -3x^2 + 9x = 54. Factoring out the greatest common factor (GCF) of 3 first, we get:

1. -3(x^2 - 3x - 18) = 0
2. -3(x - 6)(x + 3) = 0

Setting each factor equal to zero gives the roots x = 6 and x = -3.

Now factoring the equation without factoring out the GCF first, we end up with the same quadratic expression x^2 - 3x - 18 = 0 which factors out to (x - 6)(x + 3) = 0. This shows that the roots did not change and are still x = 6 and x = -3.

To find the time it will take the rider to reach the bottom of the slide, we look for the positive root since time cannot be negative in this context. Therefore, the time it will take the rider to reach the bottom of the slide is x = 6 seconds.