OG详解-OG2 数学1 Q21

The correct answer is 29​2​. According to the first equation in the given system, the value of y is -1.5. Substituting  -1.5 for y  in the second equation in the given system yields -1.5 = x² +8x + a. Adding 1.5  to both sides of this equation yields 0 = x² +8x + a +1.5. If the given system has exactly one distinct real solution, it follows that 0 = x² +8x + a +1.5 has exactly one distinct real solution. A quadratic equation in the form 0 = px² +qx + r, where p , q, and r  are constants, has exactly one distinct real solution if and only if the discriminant,  q2 -4pr , is equal to 0. The equation 0 = x² +8x + a +1.5 is in this form, where p = 1, q = 8, and r = a +1.5. Therefore, the discriminant of the equation 0 = x² +8x + a +1.5 is (8)² -4(1)(a +1.5), or 58 -4a. Setting the discriminant equal to 0  to solve for a yields 58 -4a = 0. Adding 4a to both sides of this equation yields 58 = 4a. Dividing both sides of this equation by 4 yields 58​4​= a, or 29​2​ = a. Therefore, if the given system of equations has exactly one distinct real solution, the value of a is29​2. Note that 29/2 and 14.5 are examples of ways to enter a correct answer.