Help! Discriminant help (1 Viewer)

[tonyharrison]

From HSC Maths paper 2006: Question 7 (c)(ii)
http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2006exams/pdf_doc/maths_06.pdf


(i) got you to find the discriminant of a random function, which was fine..
(ii) gave you two equations:

y = 2x^2 + kx + 9
y = 2x + 1

and told you to find the values for k for which these two functions don't intersect.
I've tried to solve them simultaneously, and come up with an equation that is the function from (i), however i am not sure of the significance of this and have no idea how to find k.
Any help would be much appreciated. Thanks!!

 

[annabackwards]

Equate the equations so you get
2x^2 + kx + 9 = 2x + 1
2x^2 + (k -2)x + 8 = 0

Discrimant = (k-2)^2 - 4(2)(8) < 0 for no roots IE no solutions to the simultaneous; to find where the functions don't interact.

(k^2 - 4k + 4) - 64 < 0

k^2 - 4k - 60 < 0
(k - 10) (k + 6 ) < 0

Graphing, you'll find that the values of k are -6< k<10

 

[lyounamu]

From HSC Maths paper 2006: Question 7 (c)(ii)
http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2006exams/pdf_doc/maths_06.pdf


(i) got you to find the discriminant of a random function, which was fine..
(ii) gave you two equations:

y = 2x^2 + kx + 9
y = 2x + 1

and told you to find the values for k for which these two functions don't intersect.
I've tried to solve them simultaneously, and come up with an equation that is the function from (i), however i am not sure of the significance of this and have no idea how to find k.
Any help would be much appreciated. Thanks!!

The equation that incorporates both functions appear like this:2x+1 = 2x^2 + kx + 9 2x^2 + (k-2)x + 8 = 0Since they don't intersect, discrimninant is less than 0. i.e. discriminant = (k-2)^2 - 4 x 2 x 8 < 0 k^2 - 4k + 4 - 64 < 0k^2 - 4k - 60 < 0 (k-10)(k+6) < 0 therefore, -6<k<10

 

[tonyharrison]

The equation that incorporates both functions appear like this:2x+1 = 2x^2 + kx + 9 2x^2 + (k-2)x + 8 = 0Since they don't intersect, discrimninant is less than 0. i.e. discriminant = (k-2)^2 - 4 x 2 x 8 < 0 k^2 - 4k + 4 - 64 < 0k^2 - 4k - 60 < 0 (k-10)(k+6) < 0 therefore, -6<k<10

How do you know that they don't intersect if you don't know wat k is?

 

[untouchablecuz]

How do you know that they don't intersect if you don't know wat k is?

whether or not they intersect is completely dependent on the value of k
the Q isnt saying that the curves will ALWAYS intersect
rather, it is asking you to find the values for k for which they do intersect
we know that if they dont intersect, discriminant<0
so we use this to find a restriction on k

 

[tonyharrison]

whether or not they intersect is completely dependent on the value of k
the Q isnt saying that the curves will ALWAYS intersect
rather, it is asking you to find the values for k for which they do intersect
we know that if they dont intersect, discriminant<0
so we use this to find a restriction on k

Ok thankyou, that makes sense.
Just one quick question. So even though usually the discriminant is used to find the roots on the x-axis, if you solve two equations simultaneously, the roots of that equation are where they intersect?

 

[Cloesd]

From HSC Maths paper 2006: Question 7 (c)(ii)
http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2006exams/pdf_doc/maths_06.pdf


(i) got you to find the discriminant of a random function, which was fine..
(ii) gave you two equations:

y = 2x^2 + kx + 9
y = 2x + 1

and told you to find the values for k for which these two functions don't intersect.
I've tried to solve them simultaneously, and come up with an equation that is the function from (i), however i am not sure of the significance of this and have no idea how to find k.
Any help would be much appreciated. Thanks!!

The main point to notice here is that you can use the discriminant not just to find when a parabola touches an axis line, but also when in (or when not) it touches another line, by equating them with eachother, then using the discriminant processes onto the resulting line.