Maximize the roots of this quadratic equation.

Algebra Level 2

Let a, b, and c be three distinct one-digit integer numbers. What is the maximum value of the sum of the roots of the equation (x-a)(x-b) + (x-b)(x-c) = 0 ?


The answer is 16.5.

Vijay Simha by Vijay Simha , 22, USA

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2 solutions

Sum of the roots of the equation is (a+2b+c)/2. For this to be the maximum, b must be the maximum possible, which is 9. a and c must be the next highest, i.e. 8 and 7. Therefore the maximum value of (a+2b+c)/2 is (15+18)/2 or 16.5

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Mohammed Imran
Mar 28, 2020

This equation can factored into ( x − b ) ( 2 x − ( a + c ) ) (x-b)(2x-(a+c)) . So the 2 roots are x = b , a + c 2 x=b,\frac{a+c}{2} . So, max will be 9 + 7.5 = 16.5 9+7.5=\boxed{16.5}

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