The line y = x + 6 intersects the circle x^2 + y^2 = 26 at two points. What is...

1. TRANSLATE the problem information

• Given information:
  • Line equation: \(\mathrm{y = x + 6}\)
  • Circle equation: \(\mathrm{x^2 + y^2 = 26}\)
  • Need: greatest x-coordinate where they intersect
• This tells us we need to find where both equations are satisfied simultaneously

2. INFER the solution approach

• Since we have y expressed in terms of x from the line equation, we can substitute this into the circle equation
• This will give us a quadratic equation in x only

3. SIMPLIFY by substitution and expansion

• Substitute \(\mathrm{y = x + 6}\) into \(\mathrm{x^2 + y^2 = 26}\):
  \(\mathrm{x^2 + (x + 6)^2 = 26}\)
• Expand \(\mathrm{(x + 6)^2}\):
  \(\mathrm{x^2 + x^2 + 12x + 36 = 26}\)

4. SIMPLIFY further by combining and rearranging

• Combine like terms:
  \(\mathrm{2x^2 + 12x + 36 = 26}\)
• Subtract 26 from both sides:
  \(\mathrm{2x^2 + 12x + 10 = 0}\)
• Divide by 2:
  \(\mathrm{x^2 + 6x + 5 = 0}\)

5. SIMPLIFY by factoring

• Factor the quadratic:
  \(\mathrm{(x + 1)(x + 5) = 0}\)
• This gives us:
  \(\mathrm{x = -1}\) or \(\mathrm{x = -5}\)

6. CONSIDER ALL CASES to find the answer

• We found two x-coordinates: -1 and -5
• The question asks for the greatest value
• Since \(\mathrm{-1 \gt -5}\), the greatest x-coordinate is -1

Answer: A) -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x + 6)^2}\) or combining terms, leading to an incorrect quadratic equation.

Many students incorrectly expand \(\mathrm{(x + 6)^2}\) as \(\mathrm{x^2 + 36}\) (forgetting the middle term \(\mathrm{12x}\)), which would give \(\mathrm{x^2 + x^2 + 36 = 26}\), leading to \(\mathrm{2x^2 = -10}\), and \(\mathrm{x^2 = -5}\). Since this has no real solutions, this leads to confusion and guessing.

Second Most Common Error:

Poor CONSIDER ALL CASES reasoning: Students find the quadratic equation correctly but only solve for one root or select the wrong root when asked for the "greatest" value.

A student might solve \(\mathrm{(x + 1)(x + 5) = 0}\) and find \(\mathrm{x = -1}\), but then select Choice C (6) thinking the y-coordinate \(\mathrm{y = x + 6 = -1 + 6 = 5}\) is somehow related to the answer, or misremember which coordinate the question is asking for.

The Bottom Line:

This problem requires careful algebraic manipulation and attention to what the question is actually asking for. Students must expand correctly, factor properly, and then compare the solutions to find the greatest x-coordinate.