Thread: Algebraic Inequalities

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-x² at two distinct points.

I managed to get x² - (5 - m)x + 4 = 0 and I think i'm supposed to apply b²-4ac?

Originally Posted by erika

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-x² at two distinct points.

I managed to get x² - (5 - m)x + 4 = 0 and I think i'm supposed to apply b²-4ac?

not just b^2 - 4ac, the whole quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)]/2a

try to complete the problem using that. if you run into difficulties, get back to us

I don't get it, I can't seem to simplify it using the formula...

Originally Posted by erika

I don't get it, I can't seem to simplify it using the formula...

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?

I get it now, thank you very much =)