Problem 44 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (3 m+2)^{2}-2 m(1-m)-4 $$

Step-by-Step Solution

Answer

The simplified expression is \(11m^2 + 10m\).

Step 1: Apply square of a binomial formula

Recall the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 3m\) and \(b = 2\), so we have: \[ (3m + 2)^2 = (3m)^2 + 2(3m)(2) + (2)^2 \] Now, calculate each term: \[ (3m)^2 = 9m^2 \\ 2(3m)(2) = 12m \\ (2)^2 = 4 \] Thus, the expanded form of the expression will be: \[ 9m^2 + 12m + 4 - 2m(1 - m) - 4 \]

Step 2: Distributive Property

Now, distribute the \(-2m\) into the binomial \((1 - m)\): \[ -2m \cdot 1 = -2m \\ -2m \cdot (-m) = 2m^2 \] The expression becomes: \[ 9m^2 + 12m + 4 - 2m + 2m^2 - 4 \]

Step 3: Combine like terms

Finally, combine the like terms: \[ m^2:\, 9m^2 + 2m^2 = 11m^2 \\ m:\, 12m - 2m = 10m \\ \text{constants}:\, 4 - 4 = 0 \] So the simplified expression is: \[ 11m^2 + 10m \]

Key Concepts

Square of a Binomial

When you encounter a binomial squared, meaning you have an expression like \((a + b)^2\), it implies that you're multiplying the binomial by itself. This might seem daunting at first, but there's a handy formula that makes it much simpler:

• \((a + b)^2 = a^2 + 2ab + b^2\)

This means that you calculate the square of the first term \(a^2\), add twice the product of the two terms \(2ab\), and finally, add the square of the second term \(b^2\).

For the problem \((3m + 2)^2\), your terms are \(a = 3m\) and \(b = 2\). So, apply the formula:

• \((3m)^2 = 9m^2\)
• \(2(3m)(2) = 12m\)
• \((2)^2 = 4\)

This simplifies the expression to \(9m^2 + 12m + 4\). Understanding this formula can provide a clear path and prevent mistakes.

Distributive Property

The distributive property is a basic yet powerful tool in algebra, especially when dealing with multiplication over addition or subtraction in parentheses. It states:

• \(a(b + c) = ab + ac\)

This means that you multiply each term inside the parentheses by the term outside.

In our problem, you see \(-2m(1 - m)\). Here, \(-2m\) needs to be distributed over \(1 - m\), so:

• \(-2m \cdot 1 = -2m\)
• \(-2m \cdot (-m) = 2m^2\)

When solved, the terms become \(-2m + 2m^2\). The distributive property helps simplify the expression step by step, ensuring you don't miss out on important terms.

Combining Like Terms

Once you've expanded all parts of the equation, it's time to combine like terms for simplification. This means grouping and adding or subtracting terms that have the same variable raised to the same power.

In the simplified expression from our steps, you identify like terms:

• For \(m^2\): \(9m^2 + 2m^2 = 11m^2\)
• For \(m\): \(12m - 2m = 10m\)
• Combine constant terms, but here they cancel out: \(4 - 4 = 0\)

So, the final simplified output is \(11m^2 + 10m\).

Combining like terms is the final touch in polynomial simplification. It reduces the equation to its most concise form, ensuring clarity and accuracy in your results.