The function f is defined by \(\mathrm{f(x) = (x - 15)^2 + 10}\). If \(\mathrm{f(t) = 59}\), what is the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = (x - 15)^2 + 10}\)
  • \(\mathrm{f(t) = 59}\)
  • Need to find the sum of two possible t values

• This tells us: We need to substitute t into the function and set it equal to 59

2. TRANSLATE to create the equation

• Replace x with t and set the function equal to 59:
  \(\mathrm{(t - 15)^2 + 10 = 59}\)

3. SIMPLIFY to isolate the squared term

• Subtract 10 from both sides:
  \(\mathrm{(t - 15)^2 = 59 - 10}\)
  \(\mathrm{(t - 15)^2 = 49}\)

4. CONSIDER ALL CASES when taking the square root

• Taking the square root of both sides gives us both positive and negative solutions:
  \(\mathrm{t - 15 = \pm\sqrt{49}}\)
  \(\mathrm{t - 15 = \pm7}\)

5. SIMPLIFY to find both values of t

• This creates two separate equations:
  • When \(\mathrm{t - 15 = +7}\): \(\mathrm{t = 15 + 7 = 22}\)
  • When \(\mathrm{t - 15 = -7}\): \(\mathrm{t = 15 - 7 = 8}\)

6. TRANSLATE the final requirement

• The problem asks for the sum: \(\mathrm{22 + 8 = 30}\)

Answer: 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students take the square root but only consider the positive solution: \(\mathrm{t - 15 = 7}\), so \(\mathrm{t = 22}\).

They forget that \(\mathrm{\sqrt{49} = \pm7}\), missing the negative case entirely. This leads them to only find one value (\(\mathrm{t = 22}\)) and either submit that as their answer or become confused about what "sum of two possible values" means, leading to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when isolating the squared term, such as:

• \(\mathrm{(t - 15)^2 = 59 + 10 = 69}\) (adding instead of subtracting)
• Getting confused with the order of operations

This leads to incorrect values like \(\mathrm{t - 15 = \pm\sqrt{69}}\), resulting in non-integer answers that don't match the clean arithmetic expected in the problem.

The Bottom Line:

This problem tests whether students remember that square root operations in equation-solving contexts require considering both positive and negative solutions. The quadratic nature is disguised within a function format, making the ± requirement less obvious than in standard quadratic equations.