Question:If \(\mathrm{Q(x) = (2.3x - k)^2 - (5.1x^2 - 7)}\), and \(\mathrm{Q(x)}\) is written in the form Ax^2 + Bx...

1. TRANSLATE the problem requirements

• Given: \(\mathrm{Q(x) = (2.3x - k)^2 - (5.1x^2 - 7)}\) written as \(\mathrm{Ax^2 + Bx + C}\)
• Find: k when the coefficient of x is -9.2
• This means we need to expand Q(x) and set the x coefficient equal to -9.2

2. SIMPLIFY by expanding the binomial

• Start with \(\mathrm{(2.3x - k)^2}\):
  • Use formula \(\mathrm{(a - b)^2 = a^2 - 2ab + b^2}\)
  • \(\mathrm{(2.3x - k)^2 = (2.3x)^2 - 2(2.3x)(k) + k^2}\)
  • \(\mathrm{= 5.29x^2 - 4.6kx + k^2}\)

3. SIMPLIFY the complete expression

• \(\mathrm{Q(x) = (5.29x^2 - 4.6kx + k^2) - (5.1x^2 - 7)}\)
• Distribute the negative sign: \(\mathrm{Q(x) = 5.29x^2 - 4.6kx + k^2 - 5.1x^2 + 7}\)
• Combine like terms:
  • x² terms: \(\mathrm{5.29x^2 - 5.1x^2 = 0.19x^2}\)
  • x terms: \(\mathrm{-4.6kx}\) (only one)
  • Constants: \(\mathrm{k^2 + 7}\)

4. TRANSLATE the coefficient condition

• In standard form \(\mathrm{Q(x) = 0.19x^2 + (-4.6k)x + (k^2 + 7)}\)
• The coefficient of x is \(\mathrm{-4.6k}\)
• We're told this equals -9.2: \(\mathrm{-4.6k = -9.2}\)

5. SIMPLIFY to solve for k

• \(\mathrm{-4.6k = -9.2}\)
• \(\mathrm{k = -9.2 ÷ (-4.6) = 2.0}\)

Answer: B. 2.0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make sign errors when expanding \(\mathrm{(2.3x - k)^2}\) or when distributing the negative sign in front of \(\mathrm{(5.1x^2 - 7)}\). For instance, they might expand as \(\mathrm{5.29x^2 + 4.6kx + k^2}\) (missing the negative in the middle term) or incorrectly handle \(\mathrm{-(5.1x^2 - 7)}\) as \(\mathrm{-5.1x^2 - 7}\).

These algebraic mistakes lead to wrong coefficient identification, causing them to set up incorrect equations and get values like \(\mathrm{k = -2.0}\) or other incorrect results. This may lead them to select Choice A (-2.0).

Second Most Common Error:

Incomplete SIMPLIFY process: Students sometimes expand the binomial correctly but forget to combine like terms from both parts of the expression. They might only consider the x-coefficient from \(\mathrm{(2.3x - k)^2}\) and ignore that the second part \(\mathrm{-(5.1x^2 - 7)}\) doesn't contribute to the x-term, leading to confusion about which coefficient to use.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem requires careful algebraic manipulation with attention to signs and systematic combination of like terms. The key insight is recognizing that only the binomial expansion contributes to the x-coefficient, making the setup straightforward once the algebra is handled correctly.