If the given function \(\mathrm{f(x) = (x + 6)(x - 4)}\) is graphed in the xy-plane, where \(\mathrm{y = f(x)}\),...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = (x + 6)(x - 4)}\) graphed as \(\mathrm{y = f(x)}\)
• Find: x-coordinate of an x-intercept
• What this tells us: We need to find where the graph crosses the x-axis

2. INFER the mathematical approach

• X-intercepts occur where the graph crosses the x-axis
• At these points, the y-coordinate equals zero
• Therefore, we need to solve: \(\mathrm{f(x) = 0}\)

3. Set up the equation

• Since \(\mathrm{f(x) = (x + 6)(x - 4)}\), we have:

\(\mathrm{(x + 6)(x - 4) = 0}\)

4. INFER the solution strategy

• The function is already factored, so we can use the zero product property
• If two factors multiply to give zero, at least one factor must equal zero

5. SIMPLIFY by applying zero product property

• Either \(\mathrm{x + 6 = 0}\) OR \(\mathrm{x - 4 = 0}\)
• Solving \(\mathrm{x + 6 = 0}\): \(\mathrm{x = -6}\)
• Solving \(\mathrm{x - 4 = 0}\): \(\mathrm{x = 4}\)

Answer: -6 or 4 (either value is correct)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't connect "x-intercept" to the condition \(\mathrm{y = 0}\)

Many students understand that x-intercepts are where graphs cross the x-axis, but they struggle to translate this into the mathematical requirement that \(\mathrm{y = 0}\) (or \(\mathrm{f(x) = 0}\)). Without this key translation, they can't set up the equation properly and may attempt to expand the factored form unnecessarily or use other inappropriate methods.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when solving \(\mathrm{x + 6 = 0}\)

Students correctly set up \(\mathrm{(x + 6)(x - 4) = 0}\) and apply zero product property, but when solving \(\mathrm{x + 6 = 0}\), they incorrectly get \(\mathrm{x = 6}\) instead of \(\mathrm{x = -6}\). This happens because they forget that subtracting 6 from both sides gives \(\mathrm{x = 0 - 6 = -6}\).

This leads them to provide only one correct answer (4) while missing the other (-6).

The Bottom Line:

Success on this problem requires recognizing that "x-intercept" translates to "where \(\mathrm{y = 0}\)" and then carefully executing basic algebraic steps without sign errors.