An algebra problem by Arnab Das

Algebra Level 3

Consider a quadratic equation F ( x ) = 0 F(x)=0 , which attains a maximum value of 9 at x = 2.5. Then the sum of roots of the equation F ( x ) + 25 = 0 F(x) +25=0 is


The answer is 5.

Arnab Das by Arnab Das , 24, India

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1 solution

Jyotsna Bapat
Feb 5, 2015

Let F(x)=x^2+ax+b. Using two conditions given (i.e. F(5/2)=9 and F'(5/2)=0) we get F(x)=x^2-5x+25/4 F(x)=25=x^2-5x+125/4. Roots are 25/4 and -5/4 so the sum is 20/4=5

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You dont need to solve for the whole quadratic. Since you are adding just a constant, the product of roots will change but the sum of roots will not change. At x = -b/2a [For ax^2 + bx + c = g(x)], we get the minimum/maximum value depending upon the sign of the coefficient of x^2. Sum of roots of g(x) is S = -b/a. Therefore, minimum/maximum value occurs at x = S/2. Since here maximum value occurs at x = 2.5, we can say that S/2 = 2.5. Therefore, S = sum of roots of F(x) = Sum of roots of F(x) + 25 = 5.

Akshay Mujumdar - 6 years, 4 months ago

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