If the equations $x^2+bx+c=0$ have root of opposite sign of the roots of the equation $x^2-5x+6=0$ then find the values of $a \ and\ b$?

The roots of the equation x^2 - 5x + 6 = 0 are found by factoring into (x - 2)(x - 3) = 0, giving roots of 2 and 3. The new equation must have roots with the opposite sign, meaning the roots are -2 and -3. For a new quadratic equation x^2 plus bx plus c = 0, the sum of the new roots is (-2) plus (-3) = -5, so b equals 5. The product of the new roots is (-2) times (-3) = 6, but because the standard form uses a plus sign, c must be -6, making the values b = 5 and c = -6.

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