OG详解-OG1 数学2 Q27

The correct answer is 289. A quadratic equation of the form  ax² + bx +c = 0, where a , b, and c are constants, has no real solutions when the value of the discriminant, b² -4ac, is less than 0. In the given equation,  x²-34x +c = 0, a = 1 and b =-34. Therefore, the discriminant of the given equation can be expressed as (-34)² -4(1)(c), or 1,156-4c. It follows that the given equation has no real solutions when 1,156-4c <0. Adding 4c to both sides of this inequality yields 1,156 < 4c. Dividing both sides of this inequality by 4 yields  289 < c , or c >289 . It’s given that the equation x²-34x +c = 0 has no real solutions when c > n. Therefore, the least possible value of n is 289.