Question

4. What is the possible discriminant of the graph?

a parabola opening up with vertex near negative 1.2 comma negative 9.1. a y intercept of negative 6

-11
zero
25
73

5. What is the possible discriminant of the graph?

a parabola opening up x intercepts near negative 7 and negative 3

-13
zero
15
16

Answer 1

http://www.regentsprep.org/Regents/math/algtrig/ATE3/discriminant.htm

Answer 2

@Abhisar do you get this?

Answer 3

@mathmate @Hero

Answer 4

I'll take a look at that.
It's possible to calculate, but only approximately, because the given information is "near"...

Answer 5

ok! I would appreciate it much! :)

Answer 6

@mathmate

Answer 7

Sorry, I was busy with other ongoing replies. Will be working on it soon.

Answer 8

thx

Answer 9

Finally back!

Answer 10

Recall the quadratic formula:
\(\huge x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\)

Answer 11

If we consider
\(\Delta=b^2-4ac \)
then
\(\huge x=\frac{-b\pm \sqrt{\Delta}}{2a}\)
and consequently
\(\huge x_2-x_1= \frac{\sqrt{\Delta}}{a}\)
We can read off \(x_2-x_1\) from the graph.
Can you figure out how you can find "a" to complete the solution?

Answer 12

@superhelp101
AFK for a few minutes, brb.

Answer 13

I don't really understand

Answer 14

Are you ok till the quadratic formula?

Answer 15

yes

Answer 16

Subsequently, I have replaced the discriminant by \(\Delta\) to focus on the problem.

Answer 17

ok up to that point?

Answer 18

@superhelp101 still there?

Answer 19

yep

Answer 20

did you get 73 for the #4?

Answer 21

I have a hard time reading the two zeroes, so cannot say.
Can you tell me what the two zeroes are, or better still, what is x2-x1?

Answer 22

(-3.306,0) and (.806, 0) 4.112 is the difference

Answer 23

I not sure if that's exactly what you asked?

Answer 24

Yes, 4.112 is what I need.
I have a=2.15 approx.
So according to the previous formula,
\(x_2-x_1=\sqrt{\Delta}/a\)
so
\(\Delta = (a(x_2-x_1))^2=a^2(x_2-x_1)^2\)
=78 approx.

Answer 25

so do you think 73 makes sense

Answer 26

I rechecked a=2.1528, can you recheck (x2-x1).
I don't think there should be that big a difference (78-73)/73=7%.

Answer 27

but I did .806-(-3.306) and got 4.112

Answer 28

Thank you. I will do some rechecking and get back to you.

Answer 29

ok. will it take long thou lol?

Answer 30

No, actually I just finished checking.

Answer 31

ok

Answer 32

I'll try a different but hopefully better approach.

Answer 33

If the roots are given to 3 decimal places, we'd better use them.

Answer 34

The answer \(should\) be more accurate.

Answer 35

We write
f(x)=a(x-x1)(x-x2)
Since f(0)=-6 (y-intercept), we get
a=-6/f(0)=-6/(x1*x2)=2.252 approximately.

Answer 36

ok so far?
Please check my numbers.

Answer 37

yes just checked.

Answer 38

So next we expand f(x) above numerically to get:

Answer 39

\(f(x)=2.252x^2+5.629x-6\)
from which we can easily calculate the discriminant \(b^2-4ac\).

Answer 40

31.685641-4(2.252)(-6)

Answer 41

31.685641-54.048

Answer 42

Haha, this is even bigger.
Since some of the numbers are approximate, perhaps the answers are based on some calculations of the approximate numbers. We had 78 before using those numbers.
We cannot go wrong with what we've got, because they are accurate to 3 figures!

Answer 43

The only approximate number we used was f(0)=y-intercept = -6.

Answer 44

what should we do now?

Answer 45

I would take 73.
If they say it's not correct, you can show your prof what we've done, and let him explain.
The only catch is if your prof showed you a different method using the approximate values (h,k) and f(0), which \(might\) give a different answer.

Answer 46

ok thxx!!

Answer 47

You're welcome! :)