Stationary points question?

Question

I can't find anything on mathsnet.net or other maths sites. I have a past paper question to do:

the curve C has equation

y=4x^2-7x+11

and the line L has equation

y=5x+k

where k is a constant. Given that L intersects C in two distinct points, show that k>2.

I'd appreciate help on understanding the question more than an answer thanks. :D

The question is worth 6 marks.

Answer 1

the curve C has equation

y=4x^2-7x+11

and the line L has equation

y=5x+k

where k is a constant. Given that L intersects C in two distinct points, show that k>2.

if the curve and the line intersect then:
4x^2-7x+11 =5x+k should have 2 solutions,
4x^2 -12x + 11-k=0
x=( 12 +- sqrt( 12^2 -4(4)(ll-k) ) /2(4)
this has two solutions if
12^2 -4(4)(ll-k) >0
9-(11-k) >0
-2+k >0
so k>2 .

Answer 2

put the two eq. equal to each other to solve for the pts of intersection:
4x^2 -7x +11=5x +k
rearrange:
4x^2 -12x +(11-k)=0
for L to intersect C in two distinct points
the dicriminant >0(from the quadratic formula)
i.e. b^2 - 4ac>0
(-12)^2 -4(4)(11-k)>0
144-176+16k>0
16k>32
k>2