Factoring Quadratics Find The Missing Factor Of Y^2 - 10y + 24

At the heart of algebra lies the concept of factoring, a fundamental skill that unlocks the door to solving equations, simplifying expressions, and understanding the behavior of functions. Factoring quadratic expressions is a particularly important technique, and this article delves into the process with a specific example. We'll explore how to find the missing factor of a quadratic expression when one factor is already known. This exploration isn't just an academic exercise; it's a practical tool that forms the foundation for more advanced mathematical concepts.

To truly understand the mechanics of factoring, let's first dissect what a quadratic expression is. A quadratic expression is a polynomial expression of degree two, meaning the highest power of the variable is two. The general form of a quadratic expression is ax^2 + bx + c, where a, b, and c are constants, and x is the variable. Our focus in this article is on quadratic expressions where a equals 1, which simplifies the form to x^2 + bx + c. Factoring such expressions involves breaking them down into the product of two binomials, each containing a variable term and a constant term. This reverse engineering of the expansion process allows us to identify the roots of the quadratic equation, which are the values of the variable that make the expression equal to zero.

The method we'll employ hinges on the relationship between the coefficients of the quadratic expression and the constants within the binomial factors. When we expand the product of two binomials, say (x + p)(x + q), we obtain x^2 + (p + q)x + pq. Comparing this to the general form x^2 + bx + c, we observe that the coefficient b is the sum of p and q, while the constant term c is the product of p and q. This connection provides a powerful strategy for factoring: find two numbers that add up to b and multiply to c. These numbers then become the constant terms in our binomial factors. This method, while straightforward, requires careful attention to signs and a systematic approach to identifying the correct number pairs.

Let's consider the specific problem at hand: the expression y^2 - 10y + 24 has a factor of (y - 4). Our mission is to find the other factor. This problem provides an excellent opportunity to apply the factoring principles we discussed earlier. We know one of the binomial factors, and we need to determine the remaining one. The challenge lies in using the given information to deduce the unknown factor. This type of problem is not just about finding the answer; it's about understanding the underlying structure of quadratic expressions and how their factors relate to the coefficients.

To solve this problem, we can leverage the relationship between the factors and the coefficients of the quadratic expression. We know that when we multiply the two factors, we should get back the original expression, y^2 - 10y + 24. Since we already know one factor is (y - 4), we can set up an equation to represent this relationship. Let the other factor be (y + k), where k is a constant that we need to determine. The equation becomes (y - 4)(y + k) = y^2 - 10y + 24. This equation is the key to unlocking the solution. By expanding the left side and comparing coefficients with the right side, we can solve for k. This process highlights the power of algebraic manipulation in solving factoring problems.

Expanding the left side of the equation (y - 4)(y + k) = y^2 - 10y + 24, we get y^2 + ky - 4y - 4k. Combining like terms, this simplifies to y^2 + (k - 4)y - 4k. Now, we have an expression that is equal to y^2 - 10y + 24. For these two expressions to be equal, the coefficients of the corresponding terms must be equal. This gives us two equations: k - 4 = -10 and -4k = 24. These equations are derived from the coefficients of the y term and the constant term, respectively. Solving these equations will lead us to the value of k, which will reveal the other factor of the quadratic expression. This step-by-step approach demonstrates how a complex problem can be broken down into smaller, manageable parts.

Let's embark on the journey of finding the missing factor in a step-by-step manner. Our quadratic expression is y^2 - 10y + 24, and we know one of its factors is (y - 4). Our goal is to find the other factor. To achieve this, we'll employ the factoring principles discussed earlier, utilizing the relationship between the factors and the coefficients of the quadratic expression. This step-by-step approach will not only lead us to the correct answer but also solidify our understanding of the factoring process.

Step 1: Assume the Other Factor

We'll assume the other factor has the form (y + k), where k is a constant that we need to determine. This assumption is based on the understanding that the quadratic expression can be factored into two binomials. The variable term in each binomial will be y since the leading coefficient of the quadratic expression is 1. The constant term k is what we need to find. This step sets the stage for the subsequent steps, where we'll use algebraic manipulation to solve for k.

Step 2: Set Up the Equation

Knowing one factor is (y - 4) and assuming the other is (y + k), we can set up the equation (y - 4)(y + k) = y^2 - 10y + 24. This equation represents the relationship between the factors and the original quadratic expression. It's a crucial step because it translates the factoring problem into an algebraic equation that we can solve. The left side of the equation represents the product of the two factors, while the right side is the original quadratic expression. Solving this equation for k will reveal the missing factor.

Step 3: Expand the Left Side

Expanding the left side of the equation, (y - 4)(y + k), we use the distributive property (often referred to as the FOIL method) to multiply each term in the first binomial by each term in the second binomial. This gives us y^2 + ky - 4y - 4k. Expanding the product is a key step in relating the unknown constant k to the known coefficients of the quadratic expression. This expansion allows us to compare the coefficients and set up equations to solve for k.

Step 4: Simplify the Expanded Expression

Combining like terms in the expanded expression y^2 + ky - 4y - 4k, we get y^2 + (k - 4)y - 4k. This simplification makes it easier to compare the coefficients with the original quadratic expression. The simplified expression now clearly shows the coefficient of the y term as (k - 4) and the constant term as -4k. This step prepares us for the next step, where we'll equate the coefficients and solve for k.

Step 5: Equate Coefficients

Now we have y^2 + (k - 4)y - 4k = y^2 - 10y + 24. For these two expressions to be equal, the coefficients of the corresponding terms must be equal. This gives us two equations:

• k - 4 = -10 (equating the coefficients of the y term)
• -4k = 24 (equating the constant terms)

These equations provide a pathway to solve for k. By setting the coefficients equal, we transform the problem of finding a factor into a system of linear equations. Solving these equations will reveal the value of k, which in turn will give us the other factor.

Step 6: Solve for k

Let's solve the equations we obtained in the previous step.

• From k - 4 = -10, we add 4 to both sides to get k = -6.
• From -4k = 24, we divide both sides by -4 to get k = -6.

Both equations yield the same result, k = -6. This consistency confirms that our approach is correct and that we've found the correct value for k. This step is the culmination of our algebraic manipulation, providing us with the key to unlocking the other factor.

Step 7: Determine the Other Factor

Since we assumed the other factor to be (y + k) and we found k = -6, the other factor is (y - 6). This is the missing piece of the puzzle. We've successfully factored the quadratic expression y^2 - 10y + 24 into (y - 4)(y - 6). This final step demonstrates the power of the factoring process and how it allows us to break down complex expressions into simpler components.

Therefore, the other factor of y^2 - 10y + 24 is (y - 6), which corresponds to option D. But, we don't stop here. It is a great idea to always verify our answer to ensure accuracy. To verify, we can multiply the two factors we found, (y - 4) and (y - 6), and see if we get back the original expression. This verification step is a crucial part of the problem-solving process, ensuring that we haven't made any mistakes along the way.

Multiplying (y - 4)(y - 6), we get y^2 - 6y - 4y + 24, which simplifies to y^2 - 10y + 24. This matches the original expression, confirming that our answer is correct. Verification not only confirms the answer but also reinforces our understanding of the factoring process. It demonstrates the reversibility of factoring and expansion, highlighting the connection between these two operations.

In conclusion, we've successfully found the other factor of the quadratic expression y^2 - 10y + 24, given that one factor is (y - 4). The other factor is (y - 6). This problem illustrates the power of factoring quadratic expressions, a fundamental skill in algebra. By understanding the relationship between the factors and the coefficients, we can efficiently solve problems like this. Factoring is not just a mathematical technique; it's a way of thinking about expressions and equations, breaking them down into simpler components to reveal their underlying structure.

The step-by-step solution we've walked through highlights the importance of a systematic approach to problem-solving. From assuming the form of the other factor to setting up and solving equations, each step builds upon the previous one, leading us to the correct answer. The verification step further reinforces the importance of accuracy and confirms our understanding of the concepts involved. Mastering quadratic factoring is essential for success in algebra and beyond. It provides a foundation for solving quadratic equations, simplifying algebraic expressions, and tackling more advanced mathematical concepts. The skills and techniques we've explored in this article will serve you well in your mathematical journey.

The correct answer is D. y - 6