y = - 1.50 y = x 2 + 8 x + a In the given system of equations, a

Question

y = - 1.50 y = x 2 + 8 x + a In the given system of equations, a is a positive constant. The system has exactly one distinct real solution. What is the value of a ?

Accepted Answer

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

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