1. Oct 4, 2006

physicsgal

it's an equation about a football player kicking a football..
h = -4.9t^2 + 10t + 3
h = height (in meters)
t = time (in seconds)

so the vertex is 9.25m height at 1.25 seconds? (i figured this on my graphing calculator).

and the other question asks how many seconds til the ball reaches the ground and by using the quadratic formula i got 2.3 seconds.

then the last question has to do about how much time is the ball above the 5 meter mark.

so this would be:
5m = -4.9t^2 + 10t + 3?
i dunno how i'd figure this out. any help is appreciated!

~Amy

2. Oct 4, 2006

The vertex is $$-\frac{b}{2a}$$. So its $$(1.02, 8.10)$$. Your second answer is correct.

For the third one, you do the following

$$-4.9t^{2} + 10t -2 = 0$$
$$t = 0.224$$

Since we know that at time 1.02 seconds the football is at its highest point, then the ball spends $$2(1.02-.224) = 1.592$$ seconds above the 5m mark.

3. Oct 4, 2006

physicsgal

im not quite seeing it yet.. where did the -2 come from? nevermind.. 3-5 = -2.. i see it now :) thanks

4. Oct 4, 2006

physicsgal

thanks for the help

here's another one that's a bit of a pickle. i've attempted it. it has to do with complex numbers.

"find the equation of a quadratic that has 3 + i and 3- i as it's roots.

= (x - x(3-i))(x - (3-i))
= x^2 - x(3-i) - x(3+i) + (3+i)(3-i)
=x^2 - 6x + 9 - i^2
x^2 - 6x + 10

does that look ok?

~Amy

5. Oct 4, 2006

yep, that is correct.

6. Oct 4, 2006

HallsofIvy

Staff Emeritus
Yes, the situation is symmetric so the time going from the 5 m height to its highest point is equal to the time it spends going back down to the level. A little more direct:
The quadratic equation $-4.9t^2+ 10t- 2= 0$ has two[\b], the smaller being when it reaches 5 m on the way up, the larger when it passes 5 m on the way down. The time it spends above 5 m is the difference between the two.

7. Oct 5, 2006

physicsgal

so to calculate any vertex it's just -B/-2A? and that gives you p or q? and then whats the easy calculation for the other one? (p or q i dunno). so the 'c' value doesnt matter?

thanks for the tips Hallsofivy

~Amy

8. Oct 5, 2006

yes $$-\frac{b}{2a}$$ is the vertex. The $$p$$ and $$q$$ you are talking about is part of the Rational Root Theorem.

$$p$$ is all the factors of the constant term.
$$q$$ is all the factors of the leading coefficient.

So you can have $$y = x^{2}-6x+7$$
$$\frac{p}{q} = \frac{{\pm1,\pm7}}{{\pm1}}$$

9. Oct 5, 2006

physicsgal

but how did you end up with two values for the vertex. like in your 2nd post in this thread you did the -b/-2a and got 1.02 and 8.10?

im not sure what you mean by the root theorem. in my lesson books +p, and q = the vertex.

y = a(x-P)^2 + q

~Amy

10. Oct 5, 2006

8.10 is the y-coordinate of the vertex. I substituted in the 1.02 into $$h = -4.9t^2 + 10t + 3$$.

So $$y = a(x-1.02)^{2} + 8.10$$

Last edited: Oct 5, 2006
11. Oct 5, 2006

physicsgal

i see now

so according to that x (of the vertex) = p, and y(of the vertex) = q?

~Amy

12. Oct 5, 2006

yes, that is correct

13. Oct 5, 2006

physicsgal

thanks, that makes things a lot easier for me

~Amy

14. Oct 5, 2006

drpizza

Hopefully you still check back to this thread..
Here's a trick to avoid multiplying a trinomial times a trinomial on such problems..

Once you simplify the (x-(3+i))(x-(3-1)), you get
(x-3-i)(x-3+i) Notice that the x-3 is the same in both factors.
Draw a box around the x-3 in both of the factors. Ignoring what's in the box (but it's the same in both boxes), it looks like
(box-i)(box+i) which you (hopefully) recognize is a conjugate pair.

So, the product is (box)^2 - i^2

Since you'll probably be faced with a lot of problems in which either the complex roots or the irrational roots will be a conjugate pair of roots, writing the factors as a conjugate pair in this manner is a great time saver. I avoided showing it more formally, choosing to have you draw a box around the x-3's, rather than a set of parenthesis, simply because that way has "worked" better for students I've had.

15. Oct 11, 2006

physicsgal

i see what you're saying and i wrote it in my study notes/cheat sheet . i dont have much in the way of mathematical aptitude, so any quick tricks help

thanks

~Amy