Question

Find the product. $$ (9.4 y-5.1)(7.3 y-12.2) $$

Short answer

The product of the given binomials is \(68.62y^2 - 151.91y + 62.22\).

Step-by-step solution

Apply the FOIL Method

First, distribute the first factors in both binomials: \(9.4y \times 7.3y\). Next, calculate the outer factors: \(9.4y \times -12.2\). After this, find the result of the inner factors: \(-5.1 \times 7.3y\). Lastly, determine the multiplication of the last factors: \(-5.1 \times -12.2\).

Perform the Calculations

Next is to compute the four individual multiplications. This results in: \(68.62y^2\), \(-114.68y\), \(-37.23y\) and \(62.22\).

Combine Like Terms

Observe that there are two middle terms that are similar. Add them together to arrive at a simpler expression. The end result is \(68.62y^2 - 151.91y + 62.22\).

Key concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In our case, the product of two binomials we started with and the result we ended with are both algebraic expressions. These algebraic expressions play a pivotal role in algebra, allowing us to represent relationships and formulate equations we can solve.

An important thing to note is that while simplifying expressions by combining like terms, the structural integrity of the expression should remain intact. For example, variables should never be combined with constants in a sum or difference, as they are inherently different types of terms. The goal with algebraic expressions is to simplify them to a form that is easier to understand or use in further mathematical processes, which is why we use methods like FOIL and combine like terms.