help plz

The curve C has the equation y = x² + kx − 3 and the line l has the equation y = k − x,
where k is a constant.
Prove that for all real values of k, the line l will intersect the curve C at exactly two points. (7)

Reply 1

Gaz031

15

IZZY!

The curve C has the equation y = x² + kx − 3 and the line l has the equation y = k − x,
where k is a constant.
Prove that for all real values of k, the line l will intersect the curve C at exactly two points. (7)

(1) Find an equation whose solutions are the values of x at which they intersect by substitution y=k-x into the equation of the curve C.
(2) Consider the discriminant of the quadratic - prove that the discriminant is greater than 0 by completing the square.

Reply 2

Mechs

IZZY!

The curve C has the equation y = x² + kx − 3 and the line l has the equation y = k − x,
where k is a constant.
Prove that for all real values of k, the line l will intersect the curve C at exactly two points. (7)

y = x² + kx − 3
y = k − x,

k − x = x² + kx − 3
x - k + x² + kx - 3 = 0
x² + kx + x - k - 3 = 0
x² + x(k+1) - k - 3 = 0
(x + ½(k+1))² - [½(k+1)]² - k - 3 = 0
x + ½(k+1) = √[ [½(k+1)]²+k+3 ]
x = - ½(k+1) ± √[ [½(k+1)]²+k+3 ]

since the equation is a nice quadratic, the maximum solutions are 2 roots. And since when reaaranged for x, k is always postive inside the root, there are no complex roots, hence both roots will be real for all values of k.

Reply 3

Gaz031

15

You'll also need to state that since the discriminant is non zero we don't have coincident roots.

Reply 4

IZZY!

OP

15

plz help me to solve this question
see attachment.

thank you

help.doc38.4KB

Reply 5

Mechs

IZZY!

plz help me to solve this question
see attachment.

y = ax² + bx + c
-6 = a(0)² + b(0) + c
c = -6
a(2)² + b(2) - 6 = 0
4a + 2b - 6 = 0
2(a + b) = 6
a + b = 3

from the graph sketch and the above:
a = -1
b = 4

: y = -x² + 4x - 6

Reply 6

Neapolitan

2

Mechs

y = ax² + bx + c
-6 = a(0)² + b(0) + c
c = -6
a(2)² + b(2) - 6 = 0
4a + 2b - 6 = 0
2(a + b) = 6
a + b = 3

from the graph sketch and the above:
a = -1
b = 4

: y = -x² + 4x - 6

a is definitvely negative from the graph but you can't say that it is -1.

c = - 6

a + b = 3

and

b/(2a) = -2

as it has repeated roots

So a = -1 and b =4 is correct.