y = − 1.5y = x2 + 8x + aIn the given system of equations, a is a positive constant. The system has exactly one distinct real solution. What is the value of a ?

Explanation

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 2 - =++ 1.5 8 x x a. Adding 1.5 to both sides of this equation yields 2 0 8 1.5 = + ++ x x a . If the given system has exactly one distinct real solution, it follows that 2 0 8 1.5 = + ++ x xa has exactly one distinct real solution. A quadratic equation in the form 2 0= ++ px qx r, where p, q, and r are constants, has exactly one distinct real solution if and only if the discriminant, 2 q pr -4 , is equal to 0. The equation 2 0 8 1.5 = + ++ x xa is in this form, where p =1, q =8, andr = a+1.5. Therefore, the discriminant of the equation 2 0 8 1.5 = + ++ x xa is 28 4 1 1.5 - +a , or 58 4 - a. Setting the discriminant equal to 0 to solve for a yields 58 4 0 - =a . Adding 4a to both sides of this equation yields 58= 4 a.Dividing both sides of this equation by 4 yields 584 = a, or 292 = a. Therefore, if the given system of equations has exactly one distinct real solution, the value of a is 29 /2 . Note that 29/2 and 14.5 are examples of ways to enter a correct answer.