Question

Let `alpha` and `beta` be the roots of the equation `(k-1)x^2+(1-5k)x+4k=0` where k is real and k does not equal to 1. Find the value of k if `alpha=beta` ?

Answer 1

`k=-1/3`

Explanation:

Let α and β be the roots of the equation
`(k−1)x²+(1−5k)x+4k=0`, where k is real and k does not equal to 1. Find the value of k if `α=β`
Well, you don't have to worry actually: you just need to understand what you need to do.
What are α and β related to the equation ? They are the solutions, sure. So, if we have `α=β`, it means that `Δ=0`
Let `Δ=b²-4ac`

`Δ=(1-5k)²-4*4k(k-1)`

`Δ=25k²-10k+1-16k²+16k`

`Δ=9k²+6k+1`

And also `Δ=0`

So: `9k²+6k+1=0`

We have a second equation but now our unknown value is k.
Let `δ=b²-4ac`

`δ=6²-4*9*1`

`δ=36-36`

`δ=0`

So because `δ=0`, there is an unic solution for this second equation.
`k=-b/(2a)`
`k=-6/18`
`k=-1/3`
\0/ here's our answer!