- x 2 + b x - 676 = 0 In the given equation, b is a positive integer. The

Question

- x 2 + b x - 676 = 0 In the given equation, b is a positive integer. The equation has no real solution. What is the greatest possible value of b ?

Accepted Answer

The correct answer is 51 . A quadratic equation of the form a x 2 + b x + c = 0 , where a , b , and c are constants, has no real solution if and only if its discriminant, - 4 a c + b 2 , is negative. In the given equation, a = - 1 and c = - 676 . Substituting - 1 for a and - 676 for c in this expression yields a discriminant of b 2 - 4 - 1 - 676 , or b 2 - 2,704 . Since this value must be negative, b 2 - 2,704 < 0 , or b 2 < 2,704 . Taking the positive square root of each side of this inequality yields b < 52 . Since b is a positive integer, and the greatest integer less than 52 is 51 , the greatest possible value of b is 51 .

Question x-2-b-