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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Hard

$y = - 1.50$

$y = x^{2} + 8 x + a$

In the given system of equations, $a$ is a positive constant. The system has exactly one distinct real solution. What is the value of $a$ ?

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Explanation

The correct answer is $StartFraction 29 Over 2 EndFraction$ . According to the first equation in the given system, the value of $y$ is $negative 1.5$ . Substituting $negative 1.5$ 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, $q squared minus 4 p r$ , is equal to $0$ . The equation $0 = x^{2} + 8 x + a + 1.5$ is in this form, where $p equals 1$ , $q equals 8$ , 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 $58 minus 4 a$ . 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 $58 equals 4 a$ . Dividing both sides of this equation by $4$ yields $\frac{58}{4} = a$ , or $\frac{29}{2} = a$ . Therefore, if the given system of equations has exactly one distinct real solution, the value of $a$ is $StartFraction 29 Over 2 EndFraction$ . Note that 29/2 and 14.5 are examples of ways to enter a correct answer.