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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Hard

$- x^{2} + b x - 676 = 0$

In the given equation, $b$ is a positive integer. The equation has no real solution. What is the greatest possible value of $b$ ?

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Explanation

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 $negative 1$ for $a$ and $negative 676$ for $c$ in this expression yields a discriminant of $b^{2} - 4 (- 1) (- 676)$ , or $b squared minus 2,704$ . Since this value must be negative, $b squared minus 2,704 less than 0$ , or $b squared less than 2,704$ . Taking the positive square root of each side of this inequality yields $b less than 52$ . Since $b$ is a positive integer, and the greatest integer less than $52$ is $51$ , the greatest possible value of $b$ is $51$ .