Question:The function h is defined by \(\mathrm{h(x) = \frac{(6x - 5)(x + 8)}{4}}\), and the graph of \(\mathrm{y = h(x)}\)...

1. INFER what the problem is asking

• Given: \(\mathrm{h(x) = \frac{(6x - 5)(x + 8)}{4}}\)
• Find: The x-coordinate of the positive x-intercept

• Key insight: X-intercepts occur where the graph crosses the x-axis, meaning where \(\mathrm{y = 0}\)

2. INFER the solution strategy

• To find x-intercepts, set \(\mathrm{h(x) = 0}\)
• Since \(\mathrm{h(x)}\) is a fraction, it equals zero when the numerator equals zero (and denominator ≠ 0)

3. Set up the equation

\(\mathrm{h(x) = \frac{(6x - 5)(x + 8)}{4} = 0}\)

Since \(\mathrm{4 \neq 0}\), we need:
\(\mathrm{(6x - 5)(x + 8) = 0}\)

4. INFER how to solve using zero product property

• If a product equals zero, at least one factor must equal zero
• So either \(\mathrm{6x - 5 = 0}\) OR \(\mathrm{x + 8 = 0}\)

5. SIMPLIFY each equation

From \(\mathrm{6x - 5 = 0}\):

• \(\mathrm{6x = 5}\)
• \(\mathrm{x = \frac{5}{6}}\)

From \(\mathrm{x + 8 = 0}\):

• \(\mathrm{x = -8}\)

6. APPLY CONSTRAINTS to select the final answer

• We found two x-intercepts: \(\mathrm{x = \frac{5}{6}}\) and \(\mathrm{x = -8}\)
• The problem asks for the positive x-intercept
• Since \(\mathrm{\frac{5}{6} \gt 0}\) and \(\mathrm{-8 \lt 0}\), the positive x-intercept is \(\mathrm{x = \frac{5}{6}}\)

Answer: 5/6

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual gap about x-intercepts: Some students don't immediately recognize that x-intercepts occur when \(\mathrm{y = 0}\), or they get confused about working with rational functions.

They might try to factor the entire expression or work with the denominator unnecessarily, leading to confusion and potentially guessing randomly among answer choices.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up the zero product property but make algebraic errors when solving the simple linear equations, particularly with \(\mathrm{6x - 5 = 0}\).

For example, they might solve \(\mathrm{6x - 5 = 0}\) as \(\mathrm{x = -\frac{5}{6}}\) instead of \(\mathrm{x = \frac{5}{6}}\), which could lead them to incorrectly identify \(\mathrm{-\frac{5}{6}}\) as an answer choice if available.

The Bottom Line:

This problem tests whether students understand the fundamental connection between x-intercepts and zeros of functions, combined with careful algebraic manipulation. The rational function format can intimidate students, but the key insight is that the denominator (being a non-zero constant) doesn't affect where the function equals zero.