SOLUTION: Determine the value of k in y=kx^2-5x+2 that will result in the intersection of the line y=-3x+4 with the quadratic at a) two points (1 mark) b) one points (1 mark) c) no point

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Determine the value of k in y=kx^2-5x+2 that will result in the intersection of the line y=-3x+4 with the quadratic at a) two points (1 mark) b) one points (1 mark) c) no point       Log On


   

Question 943236: Determine the value of k in y=kx^2-5x+2 that will result in the intersection of the line y=-3x+4 with the quadratic at
a) two points (1 mark)
b) one points (1 mark)
c) no point (1 mark)
My work on the question:
kx^2-5x+2=-3x+4
kx^2-2x-2=0
What I need for someone to help me with:
parts a,b and c showing and explaining how you got it and also if I did something wrong concerning the simplified form of the equations,explain where I went wrong and help recalculate the simplified form of the equations.
Thank you
Found 2 solutions by MathLover1, josgarithmetic:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

y=kx%5E2-5x%2B2 that will result in the intersection of the line y=-3x%2B4+with the quadratic at
remember that if
b%5E2+-+4ac+%3E+0 you will have two solutions
b%5E2+-+4ac+=+0 you will have 1 solution (2 equal ones)
b%5E2+-+4ac+%3C+0 , there is no solution
so apply those conditions for each of your questions
y=kx%5E2-5x%2B2++
y=-3x%2B4
-------------
kx%5E2-5x%2B2+=-3x%2B4
kx%5E2-5x%2B3x%2B2+-4=0
kx%5E2-2x+-2=0 => a=k, b=-2,+c=-2
b%5E2+-+4ac+%3E+0 you will have two solutions
%28-2%29%5E2+-+4k%28-2%29%3E+0+
4%2B+8k%3E+0+
8k%3E+-4+
+k%3E+-4%2F8+
k%3E+-1%2F2
b%5E2+-+4ac+=+0 you will have 1 solution (2 equal ones)

%28-2%29%5E2+-+4k%28-2%29=0+
4%2B+8k=+0+
8k=+-4+
k=-4%2F8+
k=-1%2F2
b%5E2+-+4ac+%3C+0 , there is no solution
%28-2%29%5E2+-+4k%28-2%29%3C0+
4%2B+8k%3C0+
8k%3C+-4
k%3C-4%2F8+
k%3C-1%2F2

y=kx%5E2-5x%2B2 for k+%3E+-1%2F2, 2 solutions ; if k=1 and 1+%3E+-1%2F2, we have
y=x%5E2-5x%2B2
then
y=x%5E2-5x%2B2
y=-3x%2B4
----------------
x%5E2-5x%2B2=-3x%2B4
x%5E2-5x%2B3x%2B2-4=0
x%5E2-2x-2=0
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-%28-2%29+%2B-+sqrt%28%28-2%29%5E2-4%2A1%2A%28-2%29+%29%29%2F%282%2A1%29+
x+=+%282+%2B-+sqrt%284%2B8+%29%29%2F2+
x+=+%282+%2B-+sqrt%2812+%29%29%2F2+
x+=+%282+%2B-+2sqrt%283+%29%29%2F2+
x+=+1%2Bsqrt%283%29
x+=+1-sqrt%283%29
if k=-1%2F2, we will have
y=x%5E2-5x%2B2%7D%7D%0D%0A%7B%7B%7By=-3x%2B4
----------------
%28-1%2F2%29x%5E2-5x%2B2=-3x%2B4
-x%5E2-10x%2B4=-6x%2B8
0=x%5E2%2B10x-6x%2B8-4
x%5E2%2B4x%2B4=0
x%5E2%2B4x%2B2%5E2=0
%28x%2B2%29%5E2=0
%28x%2B2%29=0 if x=-2 => one double solution


if k=-1, we will have
y=x%5E2-5x%2B2%7D%7D%0D%0A%7B%7B%7By=-3x%2B4
----------------
-1x%5E2-5x%2B2=-3x%2B4
-x%5E2-2x-2=0
x%5E2%2B2x%2B2=0
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-%282%29+%2B-+sqrt%28%282%29%5E2-4%2A1%2A%282%29+%29%29%2F%282%2A1%29+
x+=+%28-2+%2B-+sqrt%284-8+%29%29%2F2+
x+=+%28-2+%2B-+sqrt%28-4+%29%29%2F2+
x+=+%28-2+%2B-+2i%29%2F2+
solutions:
x+=+-1+%2B+2i+
x+=+-1+-+2i+


Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
A discriminant question. First derive the equation for the intersection, and solve for the x coordinate, which will be a formula including k.

kx%5E2-5x%2B2=-3x%2B4, simply through equating the expressions for y.
kx%5E2-2x-2=0
x=%282%2B-+sqrt%284%2B4k%2A2%29%29%2F%282k%29----solution for general quadratic equation applied
x=%282%2B-+sqrt%284%281%2B2k%29%29%29%2F%282k%29
x=%282%2B-+2%2Asqrt%281%2B2k%29%29%2F%282k%29
highlight%28x=%281%2B-+sqrt%281%2B2k%29%29%2Fk%29

Now use what you are supposed to know about the discriminant to find your answers for k, and be aware, k%3C%3E0.