In the xy-plane, the graph of y = x^2 - 5x intersects the graph of y = 2x at the...

1. INFER the approach for finding intersection points

• Key insight: Intersection points occur where both functions have the same y-value for the same x-value
• This means we need to solve: \(\mathrm{x^2 - 5x = 2x}\)

2. SIMPLIFY the equation by collecting terms

• Move all terms to one side: \(\mathrm{x^2 - 5x - 2x = 0}\)
• Combine like terms: \(\mathrm{x^2 - 7x = 0}\)

3. SIMPLIFY by factoring the quadratic

• Factor out the common factor x: \(\mathrm{x(x - 7) = 0}\)
• Apply zero product property: \(\mathrm{x = 0}\) or \(\mathrm{x = 7}\)

4. INFER the corresponding intersection points

• When \(\mathrm{x = 0}\): \(\mathrm{y = 2(0) = 0}\), so we get point \(\mathrm{(0, 0)}\)
• When \(\mathrm{x = 7}\): \(\mathrm{y = 2(7) = 14}\), so we get point \(\mathrm{(7, 14)}\)

5. INFER which point corresponds to (a, 2a)

• We're told the intersection points are \(\mathrm{(0, 0)}\) and \(\mathrm{(a, 2a)}\)
• Point \(\mathrm{(7, 14)}\) fits the form \(\mathrm{(a, 2a)}\) where \(\mathrm{a = 7}\) and \(\mathrm{2a = 14}\)

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that intersection points are found by setting equations equal, instead trying to substitute or graph individually.

Some students might try to find where each function equals zero separately, or attempt to solve each equation in isolation. This approach doesn't address the fundamental question of where the two graphs cross each other.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when collecting terms or factoring.

A common mistake is writing \(\mathrm{x^2 - 5x - 2x = x^2 - 3x = 0}\) instead of \(\mathrm{x^2 - 7x = 0}\), leading to incorrect solutions \(\mathrm{x = 0}\) or \(\mathrm{x = 3}\). This would suggest the intersection points are \(\mathrm{(0, 0)}\) and \(\mathrm{(3, 6)}\), making them think \(\mathrm{a = 3}\).

This may lead them to select an incorrect answer if 3 were among the choices.

The Bottom Line:

This problem tests whether students understand the fundamental concept that intersection points satisfy both equations simultaneously, requiring them to set the functions equal rather than solve them separately.