what is multiplied to make 3.75 but adds to be 1

Question

, completing the square

Answer 1

So if the quantities are x and y, then xy=3.75 and x+y=1, so y=1-x, and we can substitute  for y in the other equation:

x(1-x)=3.75, x-x²=3.75, x²-x+3.75=0 (same as quadratic 4x²-4x+15=0),

x²-x+0.25-0.25+3.75=0 (completing the square),

(x-0.5)²+3.50=0,

(x-0.5)²=-3.50. The solution is complex, involving imaginary i=√-1.

x-0.5=±i√3.50, x=0.5±i√3.5.

I suspect that xy=-3.75 rather than xy=3.75 so this simplifies the solution so that there is no complex solution:

x²-x+0.25-0.25-3.75=0 (completing the square),

(x-0.5)²-4=0,

(x-0.5)²=4,

x-0.5=2 or -2, so x=2.5 or -1.5. Then y=-1.5 or 2.5.

 With the correction the quadratic becomes 4x²-4x-15=0=(2x+3)(2x-5).