Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Show k>5

Two curves, y = 6 - 2x and y = k - x² cross at the points A and B. (A and B are not the same point)

Write down the quadratic equation satisfied by the x-coordinates of A and B, and hence show that k >5.

I'm not sure which way is best to show it?

6 - 2x = k - x²
x² - 2x + 6 = k
dk/dx = 2x - 2
for min, 2x - 2 = 0
x = 1
therefore min value of k = 1² - 2 x 1 + 6 = 5
Therefore k>5 as A and B must be seperate points: k=5 ⇒ one intersection

OR

x² - 2x + 6 - k = 0
(x-1)² - 1 + 6 - k = 0
(x-1)² + 5 = k
Therefore min value of k is 5

OR

x² - 2x + 6 - k = 0
b² = 4
4ac = 24 - 4k
b² > 4ac as curves must cross, b² ≠ 4ac as curves cross twice.
4 > 24 - 4k
4k > 20
k > 5

Thanks

Last edited by Daniel123 (2007-12-28 04:15:45)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Show k>5

The equation

has two distinct real roots; therefore its discriminant must be positive.

whence

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Show k>5

Ahhh ok, thanks