Question

select the correct answer.
how many real solutions exist for this system of equations?
y = -x + 1
y = -x² + 4x - 2

a. zero
b. one
c. two
d. infinite

Explanation:

Step1: Set the equations equal

Since both equal \( y \), set \( -x + 1 = -x^2 + 4x - 2 \).

Step2: Rearrange into quadratic form

Bring all terms to one side: \( x^2 - 5x + 3 = 0 \).

Step3: Calculate discriminant

For \( ax^2 + bx + c = 0 \), discriminant \( D = b^2 - 4ac \). Here, \( a = 1 \), \( b = -5 \), \( c = 3 \). So \( D = (-5)^2 - 4(1)(3) = 25 - 12 = 13 \).

Step4: Analyze discriminant

Since \( D = 13 > 0 \), the quadratic has two real roots, meaning the system has two real solutions.

Answer:

C. two