The function g is defined by \(\mathrm{g(t) = 2t^2 + 6t + \frac{13}{2}}\). What is the value of \(\mathrm{g\left(-\frac{3}{2}\right)}\)?

1. TRANSLATE the problem requirement

• Given: \(\mathrm{g(t) = 2t^2 + 6t + \frac{13}{2}}\)
• Find: \(\mathrm{g(-\frac{3}{2})}\)
• This means substitute \(\mathrm{t = -\frac{3}{2}}\) into every place where t appears in the function

2. SIMPLIFY by substituting and calculating each term

• Set up the substitution: \(\mathrm{g(-\frac{3}{2}) = 2(-\frac{3}{2})^2 + 6(-\frac{3}{2}) + \frac{13}{2}}\)

• Calculate the first term: \(\mathrm{2(-\frac{3}{2})^2}\)
  • First find \(\mathrm{(-\frac{3}{2})^2 = \frac{9}{4}}\)
  • Then multiply: \(\mathrm{2 \times \frac{9}{4} = \frac{18}{4} = \frac{9}{2}}\)

• Calculate the second term: \(\mathrm{6(-\frac{3}{2}) = -\frac{18}{2} = -9}\)

• The third term stays as: \(\mathrm{\frac{13}{2}}\)

3. SIMPLIFY by combining all terms

• Convert everything to halves: \(\mathrm{g(-\frac{3}{2}) = \frac{9}{2} + (-\frac{18}{2}) + \frac{13}{2}}\)
• Combine the numerators: \(\mathrm{g(-\frac{3}{2}) = \frac{9 - 18 + 13}{2} = \frac{4}{2} = 2}\)

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when squaring negative numbers or combining terms with different signs.

Many students incorrectly calculate \(\mathrm{(-\frac{3}{2})^2}\) as \(\mathrm{-\frac{9}{4}}\) instead of \(\mathrm{+\frac{9}{4}}\), forgetting that squaring a negative number gives a positive result. This leads to \(\mathrm{g(-\frac{3}{2}) = -\frac{9}{2} + (-9) + \frac{13}{2} = \frac{-9 - 18 + 13}{2} = \frac{-14}{2} = -7}\), causing them to select an incorrect negative answer or become confused.

Second Most Common Error:

Poor SIMPLIFY reasoning with fractions: Students struggle with fraction arithmetic when terms have different denominators or forget to convert mixed operations properly.

Some students might calculate \(\mathrm{6(-\frac{3}{2})}\) correctly as \(\mathrm{-9}\) but then incorrectly add \(\mathrm{\frac{9}{2} + (-9) + \frac{13}{2}}\) by treating \(\mathrm{-9}\) as a fraction with denominator 1, leading to computational errors and wrong final answers.

The Bottom Line:

This problem tests careful substitution combined with systematic fraction arithmetic - students who rush through the calculation or don't double-check their sign handling typically make errors that compound through the multi-step process.