Consider the expression \((-7\mathrm{y}^2 + 5) - (3 - 2\mathrm{y}^2) + 4(\mathrm{y}^2 - 2)\).The expression is equivalent to cy^2 -...

1. TRANSLATE the problem information

• Given: \((-7\mathrm{y}^2 + 5) - (3 - 2\mathrm{y}^2) + 4(\mathrm{y}^2 - 2)\) is equivalent to \(\mathrm{c}\mathrm{y}^2 - 6\)
• Need to find: The value of constant c

2. SIMPLIFY by distributing negative signs and multiplication

• Handle the subtraction first: \(-(3 - 2\mathrm{y}^2) = -3 + 2\mathrm{y}^2\)
• Expression becomes: \(-7\mathrm{y}^2 + 5 - 3 + 2\mathrm{y}^2 + 4(\mathrm{y}^2 - 2)\)
• Distribute the 4: \(4(\mathrm{y}^2 - 2) = 4\mathrm{y}^2 - 8\)
• Full expression: \(-7\mathrm{y}^2 + 5 - 3 + 2\mathrm{y}^2 + 4\mathrm{y}^2 - 8\)

3. SIMPLIFY by combining like terms

• Group y² terms: \(-7\mathrm{y}^2 + 2\mathrm{y}^2 + 4\mathrm{y}^2 = (-7 + 2 + 4)\mathrm{y}^2 = -1\mathrm{y}^2\)
• Group constants: \(5 - 3 - 8 = -6\)
• Simplified expression: \(-1\mathrm{y}^2 - 6 = -\mathrm{y}^2 - 6\)

4. TRANSLATE to find the coefficient

• Our simplified expression: \(-\mathrm{y}^2 - 6\)
• Given equivalent form: \(\mathrm{c}\mathrm{y}^2 - 6\)
• Therefore: \(\mathrm{c} = -1\)

Answer: -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when distributing the negative sign in \(-(3 - 2\mathrm{y}^2)\)

Students often write \(-(3 - 2\mathrm{y}^2) = -3 - 2\mathrm{y}^2\) instead of the correct \(-3 + 2\mathrm{y}^2\). This fundamental error with distributing negative signs leads to:

• Wrong y² coefficient calculation: \(-7 - 2 + 4 = -5\) instead of \(-7 + 2 + 4 = -1\)
• Final wrong answer of \(\mathrm{c} = -5\)

This may lead them to confusion and guessing since -5 likely isn't an expected answer.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors when combining like terms

Students correctly distribute but make calculation mistakes when adding/subtracting the coefficients or constants. For example, miscalculating \(-7 + 2 + 4\) as -1 instead of -1, or getting confused with the constant terms \(5 - 3 - 8\).

This leads to various incorrect coefficients and causes them to get stuck and guess.

The Bottom Line:

This problem tests careful algebraic manipulation with multiple opportunities for sign errors. Success requires systematic distribution and methodical combination of terms, with particular attention to negative signs.