OG详解-OG7 数学2 Q27

The correct answer is 14. It’s given by the first equation of the system of equations that y =-2.5. Substituting -2.5 for y in the second given equation, y = x² +8x+k, yields -2.5 = x² +8x+k. Adding 2.5 to both sides of this equation yields 0 = x² +8x+k+2.5. A quadratic equation of the form 0 = ax² +bx+ c, where a, b, and c are constants, has no real solutions if and only if its discriminant, b² -4ac, is negative. In the equation 0 = x² +8x+k+2.5, where k isa positive integer constant, a = 1, b = 8, and c = k+2.5. Substituting 1 for a, 8 for b, and k+2.5 for c in b² -4ac yields 8² -4(1)(k+2.5), or 64-4(k+2.5). Since this value must be negative, 64-4(k+2.5)<0. Adding 4 (k+2.5) to both sides of this inequality yields 64 < 4(k+2.5). Dividing both sides of this inequality by 4 yields 16<k+2.5. Subtracting 2.5 from both sides of this inequality yields 13.5<k. Since k is a positive integer constant, the least possible value of k is 14.