Question

question
match each function with the number of solutions it has.
drag each tile to the correct box.
tiles
n(x) = 0.5x² - 4x + 8 m(x) = 3x² + 5x + 9 p(x) = -2x² + 7
pairs
two real solutions
exactly one real solution
two complex solutions

Explanation:

To determine the number of solutions for each quadratic function \( ax^2 + bx + c \), we use the discriminant \( D = b^2 - 4ac \). The rules are:

• If \( D > 0 \): two real solutions.
• If \( D = 0 \): exactly one real solution.
• If \( D < 0 \): two complex solutions.

Step 1: Analyze \( n(x) = 0.5x^2 - 4x + 8 \)

Here, \( a = 0.5 \), \( b = -4 \), \( c = 8 \).
Calculate discriminant:
\( D = (-4)^2 - 4(0.5)(8) = 16 - 16 = 0 \).
Since \( D = 0 \), \( n(x) \) has exactly one real solution.

Step 2: Analyze \( m(x) = 3x^2 + 5x + 9 \)

Here, \( a = 3 \), \( b = 5 \), \( c = 9 \).
Calculate discriminant:
\( D = 5^2 - 4(3)(9) = 25 - 108 = -83 \).
Since \( D < 0 \), \( m(x) \) has two complex solutions.

Step 3: Analyze \( p(x) = -2x^2 + 7 \)

Rewrite as \( p(x) = -2x^2 + 0x + 7 \), so \( a = -2 \), \( b = 0 \), \( c = 7 \).
Calculate discriminant:
\( D = 0^2 - 4(-2)(7) = 0 + 56 = 56 \).
Since \( D > 0 \), \( p(x) \) has two real solutions.

Answer:

• two real solutions: \( p(x) = -2x^2 + 7 \)
• exactly one real solution: \( n(x) = 0.5x^2 - 4x + 8 \)
• two complex solutions: \( m(x) = 3x^2 + 5x + 9 \)