Question

1. determine the number of real solutions of the system \\(\

$$\begin{cases}y = x^2 + 8 \\\\ y = x + 15\\end{cases}$$

\\). a 0 b 1 c 2 d 3

Explanation:

Step1: Set equations equal

Since both equal \( y \), set \( x^2 + 8 = x + 15 \).
\( x^2 - x - 7 = 0 \)

Step2: Calculate discriminant

For quadratic \( ax^2 + bx + c = 0 \), discriminant \( D = b^2 - 4ac \).
Here, \( a = 1 \), \( b = -1 \), \( c = -7 \).
\( D = (-1)^2 - 4(1)(-7) = 1 + 28 = 29 \)

Step3: Analyze discriminant

Since \( D = 29 > 0 \), quadratic has 2 real roots. Thus, system has 2 real solutions.

Answer:

C. 2