For the given equations:
For a quadratic equation ax^2 + bx + c = 0, the roots are [-b+ sqrt (b^2 - 4ac)]/2a and [-b - sqrt (b^2 - 4ac)]/2a. The sum of the two roots is -2b / 2a = -b/a
We have the equations : x^2 + 3x - 5 = 0 and -x^2 + 3x +3 =0
The sum of the roots of x^2 + 3x - 5 = 0 is -b /a = -3
The sum of the roots of -x^2 + 3x +3 =0 is -b/a = 3
Therefore the equation -x^2 + 3x +3 =0 has the sum of its roots as 3.
We know that if ax^2 + bx + c = 0 is a quadratic equation, x1 and x2 are the roots, Then:
1. x1 + x1 = -b/a
2. x1*x2= c/a
Let us use the rule to determine the roots for both equations.
For the equation x^2 + 3x -5 = 0
==> x1+ x2 = -b/a = -3/1 = -3
For the equation -x^2 + 3x+3 = 0
==> x1+x2 = -b/a = -3/-1 = 3
Then the equation with the sum of its roots = 3 is (-x^2 +3x +3 = 0)