Question 179376
If the roots are {{{r[1]}}} and {{{r[2]}}}, I can write
{{{(x - r[1])*(x - r[2]) = 0}}}
{{{x^2 - (r[1] + r[2])x + r[1]*r[2] = 0}}}
the original equation is of the form
{{{x^2 + bx + c = 0}}}
{{{c = r[1]*r[2]}}}
(1) {{{c + 4 = (r[1])^2}}}
{{{c - 5 = r[1]*(r[1]/2)}}}
(2) {{{2*(c - 5) = (r[1])^2}}}
Subtract (1) from (2)
{{{2c - 10 - c - 4 = 0}}}
{{{c = 14}}}
The product of the roots is 14
check:
(1) {{{c + 4 = (r[1])^2}}}
{{{18 = (r[1])^2}}}
{{{r[1] = sqrt(18)}}}
{{{r[1] = 3*sqrt(2)}}}
{{{c = r[1]*r[2]}}}
{{{14 = 3*sqrt(2)* r[2]}}}
{{{r[2] = (14*sqrt(2))/6}}}
{{{r[1]*r[2] = (3*sqrt(2))*(14*sqrt(2))/6}}}
{{{r[1]*r[2] = (14*2)/2}}}
{{{14 = 14}}}
OK