SOLUTION: What value of K will cause {{{Kn^2 - 3n + 2Kn + 1}}} to have two zeroes / roots?

Question 1206552: What value of K will cause to have two zeroes / roots?
Found 4 solutions by math_tutor2020, Edwin McCravy, greenestamps, ikleyn:
Answer by math_tutor2020(3835)   (Show Source): You can put this solution on YOUR website!

Discriminant formula

Rule: If , then the quadratic will have two distinct roots.

In this case,
a = k
b = -3+2k
c = 1

It might help to think of as so you can match those terms up to the template

Let's find the expression for the discriminant.

Now consider the equation
Use the quadratic formula, which I'll leave the scratch work for the student to do, to find the following roots and Both of those decimal values are approximate.

If or then to lead back to . This will cause to have two distinct roots for variable n.

If or , then and it will make have exactly one real root for variable n.

If then it leads to and causes to have no real roots. The two roots would instead be non-real complex numbers.

Answer by Edwin McCravy(20077)   (Show Source): You can put this solution on YOUR website!

Answer by greenestamps(13327)   (Show Source): You can put this solution on YOUR website!

Exactly as posted, there are an infinite number of values of K for which the given polynomial will have two roots....

Answer by ikleyn(53751)   (Show Source): You can put this solution on YOUR website!
.

This problem is a TRAP.

The trap is in leading coefficient K at n^2.

A regular student does understand (and does know) that a quadratic equation
has two roots, when the discriminant is different from 0 (from zero),
and has one root, if the discriminant is zero.

So, it is a standard expected answer.

But here the trap starts working.

All that I said above is correct if an equation is really quadratic.

But if K= 0, the equation is just NOT a quadratic - it is linear.

In this case, the quadratic formula does not work and is not applicable.

It is what a regular student usually misses - and therefore gets a reduced score at an exam.

So, in this problem bad (or special) values of K are those numbers (real or complex),
that make the discriminant of the quadratic equation zero PLUS the value K= 0,
which makes the quadratic equation degenerated and, actually, linear.

So, this problem is a (well known) tool in hands of some examinators, when they want "to kill" some students.

It is the knowledge that good student must know in order to avoid falling into this trap.

Thus, this my solution has a very practical outcome.

RELATED QUESTIONS
Find a value of k so that 9m^2 � kn^2 will have the factors 3m + 7n and 3m � 7n (answered by stanbon)
For what value of k does x^2 + kx + 1 = 0 have two different real roots? This is what (answered by Boreal)
8x-2 ____ 5x+1 What value of x will cause the denominator of to be... (answered by solver91311)
Please help this is confusing to me. Find a value for k so tha 9m^2 - kn^2 will have the (answered by THANApHD,jim_thompson5910)
For what values of k will kx^2+kx+2=0 have two equal real... (answered by stanbon)
Find a value for k so that 9m ^2 - kn^2 will have the factors 3m+7n and 3m-7n (answered by jim_thompson5910)
find the value for k so that 9m^2-kn^2 will have the factors 3m+7n and... (answered by malakumar_kos@yahoo.com,kamallohia)
Find a value for k so that 9m^2-kn^2 will have the factors 3m + 7n and 3m... (answered by rapaljer)
For what value(s) of K will the equation 3xsquared=k-2x have real... (answered by stanbon)