Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (2 x+3 y)^{2}-(2 y+1)(3 x-2)+2(x-y) $$

Step-by-Step Solution

Answer

The simplified expression is \(4x^2 + 6xy + 9y^2 - x - y\).

Step 1: Expand the square of the binomial

To expand the square of the binomial \((2x+3y)^2\), we can use the formula \((a+b)^2 = a^2+2ab+b^2\). In this case, \(a = 2x\) and \(b = 3y\). Expanding the square: \[ (2x+3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2 \]

Step 2: Expand the product of the two binomials

Now, we expand the product of the two binomials \((2y+1)(3x-2)\). To do this, we can use the distributive property (also known as FOIL method): \[ (2y+1)(3x-2) = (2y)(3x) + (2y)(-2) + (1)(3x) + (1)(-2) \]

Step 3: Multiply the constant with the variable inside the parentheses

Next, we need to multiply the constant "2" with the variable "x-y" inside the last parentheses: \[ 2(x-y) = 2x - 2y \]

Step 4: Combine like terms

Now, we can combine all the terms from steps 1-3 and simplify: \[ (4x^2 + 12xy + 9y^2) - (6xy - 4y + 3x - 2) + (2x - 2y) \] Distribute the negative sign from the second term and combine like terms: \[ 4x^2 + 12xy + 9y^2 - 6xy + 4y - 3x + 2 + 2x - 2y \] Combine the terms: \[ 4x^2 + 6xy + 9y^2 + 4y - 3x + 2x - 2y \] Combine the remaining like terms: \[ 4x^2 + 6xy + 9y^2 - y - x \] This simplified form is the final answer: \[ 4x^2 + 6xy + 9y^2 - x - y \]

Key Concepts

Binomial Expansion

Binomial expansion is a technique used to expand expressions that have two terms raised to a power. In this exercise, we had the expression \((2x+3y)^2\), which means we need to expand a binomial squared. There is a specific formula we can apply, which is \((a+b)^2 = a^2 + 2ab + b^2\). This formula helps us quickly expand such expressions by identifying the first term \(a\) and the second term \(b\) and applying the formula.

- For this problem, \(a = 2x\) and \(b = 3y\). - Squaring the binomial gives us: \((2x)^2 + 2(2x)(3y) + (3y)^2\).
This step helps in breaking down the expression into manageable parts that are easier to handle before getting to the simplification phase.

Distributive Property

The distributive property is a fundamental algebraic property used to expand expressions containing products of terms. It's often stated as \(a(b + c) = ab + ac\). In our exercise, one of the steps required us to use this property to expand the product \((2y+1)(3x-2)\).

This is also referred to as the FOIL method when working specifically with binomials, which stands for First, Outer, Inner, Last.

• First: Multiply the first terms in each binomial: \((2y)(3x)\)
• Outer: Multiply the outer terms: \((2y)(-2)\)
• Inner: Multiply the inner terms: \((1)(3x)\)
• Last: Multiply the last terms: \((1)(-2)\)

By applying the distributive property methodically, each component of the binomial product is thoroughly expanded, leading to a clearer and more structured approach to simplifying complex algebraic expressions.

Combining Like Terms

Combining like terms is an essential skill in simplifying algebraic expressions. It involves grouping terms that share the same variables and powers. After expanding all parts of the given expression, it's crucial to simplify by adding or subtracting similar terms.

In our example: \[4x^2 + 12xy + 9y^2 - 6xy + 4y - 3x + 2 + 2x - 2y\],we can focus on combining like terms:

• Combine the \(xy\) terms: \(12xy - 6xy = 6xy\)
• Combine the \(x\) terms: \(-3x + 2x = -x\)
• Combine the \(y\) terms: \(4y - 2y = 2y\)

Finally, after all these steps, you should simplify:\[4x^2 + 6xy + 9y^2 - x - y\]. This resultant expression is far simpler, allowing you to understand and work with it more efficiently. Remember, accurate tracking of each term ensures the expression remains correct as you simplify it.