Question: A point x on the number line minimizes the sum of the squared distances to points 0 and 48,...

1. TRANSLATE the problem information

• Given information:
  • Point x on number line
  • Squared distance to point 0: \(\mathrm{x}^2\)
  • Squared distance to point 48: \(\mathrm{(x - 48)}^2\)
  • Additional fixed cost: 576
  • Want to minimize total cost

2. INFER the approach

• This is asking us to minimize a sum of squared distances plus a constant
• The total cost function will be quadratic in x
• Quadratic functions have their minimum at the vertex

3. Set up the cost function

Total cost = \(\mathrm{x}^2 + \mathrm{(x - 48)}^2 + 576\)

4. SIMPLIFY to standard quadratic form

• Expand \(\mathrm{(x - 48)}^2\): \(\mathrm{x}^2 - 96\mathrm{x} + 2304\)
• Combine terms: \(\mathrm{x}^2 + \mathrm{x}^2 - 96\mathrm{x} + 2304 + 576 = 2\mathrm{x}^2 - 96\mathrm{x} + 2880\)

5. INFER the minimization strategy

• For quadratic \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\), minimum occurs at vertex \(\mathrm{x} = -\mathrm{b}/(2\mathrm{a})\)
• Here: \(\mathrm{a} = 2, \mathrm{b} = -96\)
• \(\mathrm{x} = -(-96)/(2 \times 2) = 96/4 = 24\)

6. Calculate minimum cost

• Substitute x = 24: \(2(24)^2 - 96(24) + 2880\)
• \(= 2(576) - 2304 + 2880 = 1728\)

Answer: 1728

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert "squared distances" into mathematical expressions, especially \(\mathrm{(x - 48)}^2\). They might write the distance to 48 as just \(\mathrm{x} - 48\) instead of \(|\mathrm{x} - 48|\), or forget to square it.

This confusion in setting up the cost function leads them to work with an incorrect expression, making the entire solution invalid and causing them to guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{x}^2 + \mathrm{(x - 48)}^2 + 576\) but make algebraic errors when expanding. Common mistakes include expanding \(\mathrm{(x - 48)}^2\) as \(\mathrm{x}^2 - 48\mathrm{x} + 48^2\) instead of \(\mathrm{x}^2 - 96\mathrm{x} + 2304\), or making sign errors when combining terms.

These calculation errors lead to an incorrect quadratic form, which produces the wrong vertex location and ultimately the wrong minimum value.

The Bottom Line:

This problem tests whether students can translate a real-world optimization scenario into mathematical language and then apply quadratic minimization techniques. The key insight is recognizing that "minimizing sum of squared distances" creates a quadratic function whose minimum occurs at the vertex.