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Algebra Level 5

If the three equations ; $x^{2}+ax+12=0$ , $x^{2}+bx+15=0$ and $x^{2}+(a+b)x+36=0$ have a common root .

Then find the sum of maximum value of $a$ and minimum value of $b$ .

1 solution

Tom Engelsman
Jun 19, 2016

Upon observation, the three quadratic equations each have the common roots x = -3 or 3 as follows:

(x-3)(x-4) = x^2 - 7x + 12 = 0; (x-3)(x-5) = x^2 - 8x + 15 = 0; (x-3)(x-12) = x^2 - (7+8)*x + 36 =0;

(x+3)(x+4) = x^2 + 7x + 12 = 0; (x+3)(x+5) = x^2 + 8x + 15 = 0; (x+3)(x+12) = x^2 + (7+8)*x + 36 = 0.

Hence, the maximum value of a is 7 while the minimum for b is -8. We finally obtain the sum a + b = -1.