Hi, I've just learnt about roots of algebraic equations and have to solve this problem:
Find the range of values of m for which the line y=mx cuts the curve y=5x-4-x˛ at two distinct points.
I managed to get x˛ - (5 - m)x + 4 = 0 and I think i'm supposed to apply b˛-4ac?
we want
mx = 5x - 4 - x^2
=> x^2 + (m - 5)x + 4 = 0
by the quadratic formula:
=> x = [-(m - 5) +/- sqrt{(m - 5)^2 - 4(1)(4)}]/2
=> x = [-(m - 5) +/- sqrt{m^2 - 10m + 25 - 16}]/2
=> x = [-(m - 5) +/- sqrt{m^2 - 10m + 9}]/2
=> x = [-(m - 5) +/- sqrt{(m - 9)(m - 1)}]/2
we will have real roots as long as what is under the squareroot is nonnegative, that is greater than or equal to zero. so the range we want is:
(m - 9)(m - 1) >= 0
=> m >= 9 or m <= 1 .......i had to check the ranges on a number line to find what values work in these ranges
so the range of m we want is:
(-infinity , 1] U [9 , infinity)
still don't get it?