Question

for the equation shown, choose the description of the solutions.
42x - 4 = 4x² + 40
☑️ one real, rational
○ two real, rational
○ two real, irrational
○ two complex

how many x - intercepts does the graph of this quadratic have?
○ two x - intercepts
○ one x - intercept
○ no x - intercepts

Explanation:

To determine the number of \( x \)-intercepts of the quadratic \( y - 5 = 0.5x^2 + 8x + 3 \), we first rewrite it in standard form \( y = ax^2 + bx + c \):

Step 1: Rewrite the Equation

Simplify \( y - 5 = 0.5x^2 + 8x + 3 \) by adding 5 to both sides:
\( y = 0.5x^2 + 8x + 8 \)

Step 2: Identify \( a \), \( b \), \( c \)

For the quadratic \( y = ax^2 + bx + c \), we have:
\( a = 0.5 \), \( b = 8 \), \( c = 8 \)

Step 3: Calculate the Discriminant

The discriminant of a quadratic \( ax^2 + bx + c \) is \( D = b^2 - 4ac \). Substitute \( a = 0.5 \), \( b = 8 \), \( c = 8 \):
\[

$$\begin{align*} D &= (8)^2 - 4(0.5)(8) \\ &= 64 - 16 \\ &= 48 \end{align*}$$

\]

Step 4: Analyze the Discriminant

• If \( D > 0 \), the quadratic has two real, distinct roots (so two \( x \)-intercepts).
• If \( D = 0 \), the quadratic has one real root (a repeated root, so one \( x \)-intercept).
• If \( D < 0 \), the quadratic has two complex roots (no \( x \)-intercepts).

Since \( D = 48 > 0 \), the quadratic has two real \( x \)-intercepts.

Answer:

Two \( x \)-intercepts (the option "Two \( x \)-intercepts" from the choices).