G'day, Sujoy!
Let a&c belong to {2,4,6,8,10,12} & b belong to{22,24,26}, where a,b,&c are distinct. Find the number of equations of the form:
\(\displaystyle ax^2 +bx+c=0\), that have real roots?
For the quadratic \(\displaystyle ax^2 +bx+c=0\) to have
real roots, the
discriminant, \(\displaystyle \Delta\), must be
greater than or equal to zero.
That is,
\(\displaystyle \Delta = b^2 - 4ac \geq 0\) *
If we begin with the b=22 case:
\(\displaystyle \Delta = 22^2 - 4ac \geq 0\)
This can be rewritten:
\(\displaystyle 484 \geq 4ac\)
Divide by 4:
\(\displaystyle 121 \geq ac\)
Notice that you could choose any two distinct values for \(\displaystyle a\) and \(\displaystyle c\) fom those available and never be above 121 (the largest is \(\displaystyle 12\times10=120\)).
That is, with \(\displaystyle b=22\), \(\displaystyle ac \leq b^2\), always (from our options).
This is going to be the case when \(\displaystyle b=24\) or \(\displaystyle b=26\) also.
So as there are three values for \(\displaystyle b\),
and as there are (6 choose 2) = 15 distinct combinations of \(\displaystyle a\) and \(\displaystyle c\), there will be \(\displaystyle 3 \times 15 = 45\) possible equations.
'Tis an odd question.
* If they asked for real,
distinct (ie. different) roots then we would use a > sign instead of \(\displaystyle \geq\), but the results here would be no different.
