In the xy-plane, the graph of y = 3x^2 - 14x intersects the graph of y = x at the...

1. INFER the solution strategy

• Given information:
  • First function: \(\mathrm{y = 3x^2 - 14x}\)
  • Second function: \(\mathrm{y = x}\)
  • Known intersection points are \(\mathrm{(0, 0)}\) and \(\mathrm{(a, a)}\)
• Key insight: At intersection points, both functions have the same y-value, so we set them equal

2. TRANSLATE the intersection condition into an equation

• Since both functions equal y at intersection points:
  \(\mathrm{3x^2 - 14x = x}\)

3. SIMPLIFY to solve for x-values

• Rearrange by moving all terms to one side:
  \(\mathrm{3x^2 - 14x - x = 0}\)
  \(\mathrm{3x^2 - 15x = 0}\)
• Factor out the common factor 3x:
  \(\mathrm{3x(x - 5) = 0}\)

4. APPLY the zero product property

• Since \(\mathrm{3x(x - 5) = 0}\), either factor can equal zero:
  \(\mathrm{3x = 0}\) or \(\mathrm{x - 5 = 0}\)
  \(\mathrm{x = 0}\) or \(\mathrm{x = 5}\)

5. INFER the intersection points

• Since \(\mathrm{y = x}\) for both intersection points:
  • When \(\mathrm{x = 0}\): point \(\mathrm{(0, 0)}\) ✓
  • When \(\mathrm{x = 5}\): point \(\mathrm{(5, 5)}\)
• Therefore, \(\mathrm{a = 5}\)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when rearranging the equation \(\mathrm{3x^2 - 14x = x}\)

They might incorrectly combine terms as \(\mathrm{3x^2 - 13x = 0}\) instead of \(\mathrm{3x^2 - 15x = 0}\), or make factoring errors like \(\mathrm{3x(x - 13/3) = 0}\). This leads to wrong x-values and ultimately an incorrect value for a.

This leads to confusion and incorrect answer selection.

Second Most Common Error:

Inadequate INFER reasoning: Students don't recognize that intersection points require setting the equations equal

They might try to solve each equation individually or substitute specific values, missing the fundamental approach. Without this key insight, they cannot systematically find where the graphs intersect.

This causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students understand that graph intersections occur where functions have equal outputs, combined with solid algebraic manipulation skills. The conceptual insight drives the strategy, but execution errors in algebra derail the solution.