\((7532 + 100\mathrm{y}^2) + 10(10\mathrm{y}^2 - 110)\)The expression above can be written in the form ay^2 + b, where a...

1. TRANSLATE the problem requirements

• Given: \((7532 + 100\mathrm{y}^2) + 10(10\mathrm{y}^2 - 110)\)
• Need to: Express in form \(\mathrm{ay}^2 + \mathrm{b}\) and find \(\mathrm{a} + \mathrm{b}\)
• What this tells us: We need to simplify the expression to clearly see the coefficient of \(\mathrm{y}^2\) and the constant term

2. SIMPLIFY using the distributive property

• Apply distributive property to the second term: \(10(10\mathrm{y}^2 - 110)\)
• \(10 \times 10\mathrm{y}^2 = 100\mathrm{y}^2\)
• \(10 \times (-110) = -1100\)
• Result: \((7532 + 100\mathrm{y}^2) + (100\mathrm{y}^2 - 1100)\)

3. SIMPLIFY by combining like terms

• Remove parentheses: \(7532 + 100\mathrm{y}^2 + 100\mathrm{y}^2 - 1100\)
• Group like terms: \((100\mathrm{y}^2 + 100\mathrm{y}^2) + (7532 - 1100)\)
• Combine \(\mathrm{y}^2\) terms: \(200\mathrm{y}^2\)
• Combine constants: \(6432\)
• Final expression: \(200\mathrm{y}^2 + 6432\)

4. TRANSLATE to identify a and b values

• In the form \(\mathrm{ay}^2 + \mathrm{b}\), we have: \(\mathrm{a} = 200\) and \(\mathrm{b} = 6432\)
• Therefore: \(\mathrm{a} + \mathrm{b} = 200 + 6432 = 6632\)

Answer: 6632

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when applying the distributive property, particularly with the negative sign.

For example, they might calculate \(10(-110) = -110\) instead of \(-1100\), leading to:

\(200\mathrm{y}^2 + (7532 - 110) = 200\mathrm{y}^2 + 7422\)

This gives \(\mathrm{a} + \mathrm{b} = 200 + 7422 = 7622\), which is incorrect.

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly apply the distributive property but fail to properly combine like terms.

They might stop at \((7532 + 100\mathrm{y}^2) + (100\mathrm{y}^2 - 1100)\) without recognizing they need to collect the \(\mathrm{y}^2\) terms together to get the standard form. This leads to confusion about which values represent a and b.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful attention to sign changes during distribution and methodical combination of like terms to achieve the required form.