Question 1139198
<br>
In the quadratic equation<br>
{{{Ax^2+Bx+C = 0}}}<br>
the sum of the roots is -B/A and the product of the roots is C/A.<br>
If the roots are B and C, then<br>
(1) {{{B+C = -B/A}}}
(2) {{{BC = C/A}}}<br>
Equation (2) gives us<br>
{{{ABC = C}}}
{{{AB = 1}}}<br>
There are two possibilities with A and B both integers: they are both 1, or they are both -1.<br>
If A = B = 1 then equation (1) gives us<br>
{{{1+C = -1}}}
{{{C = -2}}}<br>
Then the equation is<br>
{{{x^2+x-2 = 0}}}<br>
{{{(x+2)(x-1) = 0}}}
{{{x = -2}}} or {{{x = 1}}}<br>
and the roots are B and C.<br>
So there is one solution to the problem.<br>
If A = B = -1, then equation (1) gives us<br>
{{{-1+C = -1}}}
{{{C = 0}}}<br>
Since the requirement is that A, B, and C be non-zero, there is no solution in this case.<br>
So the unique quadratic equation Ax^2+Bx+C=0 with roots B and C is {{{x^2+x-2 = 0}}}.<br>