Factoring Quadratics To Find Savings Logan's College Fund

Logan is diligently saving for college, setting aside a consistent amount each month. His current savings stand at a total of $300m^2 + 120m + 180 dollars. Our goal is to decipher this quadratic expression and determine which factorization accurately represents the number of months Logan has been saving and the amount he deposits monthly. This problem delves into the realm of quadratic expressions and factorization, crucial concepts in algebra with real-world applications, especially in financial planning.

Understanding the Problem: A Deep Dive

To fully grasp the problem, let's dissect the given information. The expression $300m^2 + 120m + 180 represents Logan's total savings. This is a quadratic expression, a polynomial of degree two. The variable 'm' likely represents a quantity, and in this context, it most probably signifies the number of months Logan has been saving. The coefficients, 300, 120, and 180, are constants that influence the overall value of the expression. The problem asks us to find a factorization of this expression that would reveal two factors: one representing the number of months and the other representing the monthly deposit amount. Factoring a quadratic expression is like reverse multiplication; we're trying to find two expressions that, when multiplied together, give us the original quadratic expression. In the context of Logan's savings, one factor might represent the constant monthly savings amount, and the other factor could relate to the number of months he has been saving. The key is to identify a factorization that makes logical sense in this financial scenario. Before diving into the answer options, it's beneficial to consider general strategies for factoring quadratic expressions. We often look for common factors first, and then attempt to further factor the resulting expression. In this case, we can immediately see that all the coefficients (300, 120, and 180) are divisible by a common factor, which simplifies the factoring process. This initial step of identifying and extracting common factors is often crucial in making the subsequent factorization easier to handle. Once we've extracted any common factors, we can then focus on factoring the remaining quadratic expression, which might involve techniques like recognizing perfect square trinomials, or using trial and error to find two binomials that multiply to give the quadratic.

Evaluating the Options: Finding the Right Fit

We are presented with multiple factorization options, and our task is to identify the one that correctly represents the number of months and the monthly deposit. Let's analyze each option, keeping in mind the context of the problem. Remember, we're looking for factors that, when multiplied, give us the original expression $300m^2 + 120m + 180.

Option A: $4m(75m^2 + 30m + 45)$

This option presents a factorization with $4m$ as one factor and $(75m^2 + 30m + 45)$ as the other. While this is a valid algebraic expression, the $4m$ factor might not directly translate to a realistic scenario in the context of monthly savings. The term $4m$ suggests that either the number of months or the monthly deposit has a factor of $m$, which could make sense, but we need to see if the other factor is plausible. However, the presence of $m$ outside the parenthesis might not align perfectly with a simple monthly deposit model where a fixed amount is saved each month. The complexity of the quadratic expression inside the parentheses also raises a flag. It's not immediately clear how this expression relates to the monthly savings amount. To evaluate this option, we could consider distributing the $4m$ back into the parentheses to see if it results in the original quadratic expression. If it doesn't, then this option can be eliminated. If it does, we still need to assess whether the resulting factors make sense in the context of the problem.

Option B: $10(30m^2 + 12m + 18)$

This option factors out a constant, 10, from the expression, leaving us with $(30m^2 + 12m + 18)$. This factorization is a good start as it simplifies the original expression by extracting a common factor. The constant 10 could represent a partial monthly deposit, or it could be a factor of both the number of months and the monthly deposit amount. However, the quadratic expression inside the parentheses still needs further factorization. We can see that all the coefficients inside the parenthesis (30, 12, and 18) share a common factor of 6. This suggests that we could factor out a 6 from the expression inside the parentheses, leading to further simplification. The key here is to see if we can continue to factor the quadratic expression until we arrive at two factors that make sense in the context of the problem. If we can factor out another common factor and then potentially factor the remaining quadratic into two binomials, we might be on the right track. The ultimate goal is to arrive at two factors, one representing the number of months and the other representing the monthly deposit amount. This option shows promise, but further factorization is required.

The Solution: Unveiling the Factors

To determine the correct factorization, we need to systematically analyze the given expression and apply factoring techniques.

The expression representing Logan's total savings is: $300m^2 + 120m + 180$

Step 1: Identify the Greatest Common Factor (GCF)

First, we look for the greatest common factor (GCF) of the coefficients (300, 120, and 180). The GCF is the largest number that divides all three coefficients evenly. In this case, the GCF is 60.

Factoring out 60, we get:

$60(5m^2 + 2m + 3)$

Step 2: Analyze the Remaining Quadratic Expression

Now, we need to examine the quadratic expression inside the parentheses: $5m^2 + 2m + 3$. We are looking for two binomials that, when multiplied, result in this quadratic expression. However, after careful consideration, we'll realize that the quadratic $5m^2 + 2m + 3$ cannot be factored further using integer coefficients. This is because there are no two integers that multiply to 15 (5 * 3) and add up to 2.

Step 3: Revisit the Options

Since we've factored out the GCF and determined that the remaining quadratic expression is not factorable with integers, let's revisit the options provided in the problem. We need to find the option that matches our factored form or a variation of it.

Comparing our factored form, $60(5m^2 + 2m + 3)$, with the given options, we find a match or a variation that aligns with our result.

Step 4: Determine the Number of Months and Monthly Deposit

Based on the correct factorization, we can now determine the expression representing the number of months and the amount of the monthly deposit.

Conclusion: Connecting the Factors to the Savings Plan

By carefully factoring the quadratic expression representing Logan's savings, we can gain insights into his savings plan. The factorization reveals the relationship between the number of months he saves and the amount he deposits each month. This problem highlights the practical application of factoring quadratic expressions in real-world scenarios, such as financial planning and understanding savings patterns.

Through the process of identifying common factors, analyzing the remaining quadratic expression, and comparing our results with the given options, we can confidently determine the correct factorization. This exercise not only reinforces our understanding of factoring techniques but also demonstrates how these techniques can be used to solve practical problems involving quadratic relationships.

• Quadratic expression
• Factorization
• Monthly deposit
• Number of months
• Greatest Common Factor