Problem 278 - Elementary Algebra

Question

Maribel factored \(y^{2}-30 y+81\) as \((y-9)^{2}\). Was she right or wrong? How do you know?

Step-by-Step Solution

Answer

Maribel is wrong because the expanded form does not match the original expression.

Step 1 - Identify the form

Notice that the expression given, \(y^{2}-30y+81\), is a quadratic expression of the form \(y^2 + by + c\).

Step 2 - Write the proposed factorization

According to Maribel, the factorization is \((y-9)^{2}\).

Step 3 - Expand the proposed factorization

Expand \((y-9)^{2}\) to verify the factorization: \((y-9)^{2} = (y-9)(y-9) = y^{2} - 9y - 9y + 81 = y^{2} - 18y + 81\).

Step 4 - Compare the expressions

Compare the expanded form \(y^{2} - 18y + 81\) from Step 3 with the original quadratic expression \(y^{2}-30y+81\). Notice that they are not identical because the coefficients of the middle term are different (\(-18y\) vs. \(-30y\)).

Step 5 - Conclusion

Since the expanded form does not match the original expression, Maribel's factorization is incorrect.

Key Concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power. It is particularly useful in this context for verifying the correctness of factored quadratic expressions. In binomial expansion, the following formula is often used: ewlineewlineewlineewlineewline