Solving (x+2)(x+3)=12 Using The Zero Product Property
In the realm of algebra, the zero product property stands as a cornerstone for solving polynomial equations. This property elegantly states that if the product of two or more factors is zero, then at least one of the factors must be zero. In simpler terms, if A * B = 0, then either A = 0 or B = 0 (or both). This principle is invaluable for finding the roots or solutions of equations, especially quadratic equations, which are equations of the form ax² + bx + c = 0.
Understanding the Zero Product Property
The zero product property is more than just a mathematical rule; it's a powerful tool that simplifies the process of solving equations. To truly grasp its significance, let's delve deeper into its mechanics and applications. At its core, the property hinges on the unique nature of zero in multiplication. Zero acts as an 'annihilator' in multiplication, meaning any number multiplied by zero results in zero. This characteristic is what makes the zero product property so effective. When we have an equation where a product of factors equals zero, we can confidently deduce that at least one of those factors must be zero. This transforms a complex equation into a set of simpler equations, each of which can be solved independently.
Consider the equation (x - 3)(x + 5) = 0. Here, we have two factors, (x - 3) and (x + 5), whose product is zero. According to the zero product property, either (x - 3) = 0 or (x + 5) = 0. Solving these individual equations is straightforward. For (x - 3) = 0, we add 3 to both sides, yielding x = 3. For (x + 5) = 0, we subtract 5 from both sides, resulting in x = -5. Thus, the solutions to the original equation are x = 3 and x = -5. This example illustrates the elegance and efficiency of the zero product property in breaking down a seemingly complex problem into manageable parts.
The beauty of the zero product property lies in its ability to convert multiplication problems into addition or subtraction, which are generally easier to solve. It's a fundamental concept that underpins many algebraic techniques, including factoring and solving quadratic equations. By mastering this property, students gain a crucial tool for tackling a wide range of mathematical challenges. Furthermore, the zero product property isn't confined to simple linear factors; it extends to more complex expressions and higher-degree polynomials, making it a versatile asset in any mathematical toolkit. Understanding its principles and applications is essential for anyone seeking to excel in algebra and beyond.
Applying the Zero Product Property to (x+2)(x+3)=12
Now, let's tackle the specific equation at hand: (x+2)(x+3)=12. This equation, at first glance, doesn't appear to be in the standard form required to directly apply the zero product property. The property, as we've established, works when the product of factors is equal to zero. However, in our equation, the product (x+2)(x+3) is equal to 12, not zero. This is a common hurdle in solving quadratic equations, but one that can be easily overcome with a bit of algebraic manipulation. The key is to transform the equation into a form where one side is zero, allowing us to utilize the zero product property effectively. This transformation typically involves expanding the product, simplifying the equation, and rearranging terms to achieve the desired form.
The first step in this process is to expand the left side of the equation, which involves multiplying the binomials (x+2) and (x+3). Using the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last), we multiply each term in the first binomial by each term in the second binomial. This gives us: x * x + x * 3 + 2 * x + 2 * 3, which simplifies to x² + 3x + 2x + 6. Combining like terms, we get x² + 5x + 6. So, our equation now looks like this: x² + 5x + 6 = 12. We've successfully expanded the product, but we're not quite ready to apply the zero product property yet. Remember, we need one side of the equation to be zero.
To achieve this, we subtract 12 from both sides of the equation. This ensures that the equation remains balanced while also setting the right side to zero. Subtracting 12 from both sides of x² + 5x + 6 = 12 gives us x² + 5x + 6 - 12 = 12 - 12, which simplifies to x² + 5x - 6 = 0. Now, the equation is in the standard quadratic form ax² + bx + c = 0, where a = 1, b = 5, and c = -6. This form is crucial because it allows us to factor the quadratic expression, which is the next step in applying the zero product property. Factoring involves breaking down the quadratic expression into a product of two binomials, effectively reversing the expansion process we performed earlier. Once we have the factored form, we can finally use the zero product property to find the solutions for x.
Factoring the Quadratic Expression
With our equation now in the form x² + 5x - 6 = 0, the next crucial step is to factor the quadratic expression x² + 5x - 6. Factoring is the process of breaking down a polynomial into a product of simpler polynomials, typically binomials in the case of quadratic expressions. This step is essential because it allows us to rewrite the equation in a form where the zero product property can be directly applied. The goal is to find two binomials that, when multiplied together, yield the original quadratic expression. There are several techniques for factoring, including trial and error, using the quadratic formula, and completing the square. However, for many simpler quadratic expressions, trial and error, combined with a solid understanding of factoring principles, is often the most efficient approach. In this case, we'll focus on the trial-and-error method, which involves identifying two numbers that satisfy specific criteria related to the coefficients of the quadratic expression.
To factor x² + 5x - 6, we need to find two numbers that multiply to the constant term (-6) and add up to the coefficient of the linear term (5). This is a key principle in factoring quadratic expressions of the form x² + bx + c. The two numbers we seek must have a product of -6, meaning one number must be positive and the other negative. Additionally, their sum must be 5, which gives us a clue about the relative magnitudes of the numbers. By systematically considering the factor pairs of -6, we can identify the pair that meets both criteria. The factor pairs of -6 are: (-1, 6), (1, -6), (-2, 3), and (2, -3). Among these pairs, only (-1, 6) adds up to 5. This is because -1 + 6 = 5. Therefore, the numbers we're looking for are -1 and 6.
Now that we've identified these numbers, we can rewrite the quadratic expression as a product of two binomials. The binomials will take the form (x + m)(x + n), where m and n are the numbers we found. In our case, m = -1 and n = 6. So, we can write x² + 5x - 6 as (x - 1)(x + 6). This factorization is crucial because it transforms our equation into (x - 1)(x + 6) = 0, which is perfectly suited for applying the zero product property. We've successfully broken down the quadratic expression into a product of two factors, paving the way for finding the solutions to the original equation. The next step involves using the zero product property to set each factor equal to zero and solve for x.
Applying the Zero Product Property to Find Solutions
Now that we have successfully factored the quadratic equation into the form (x - 1)(x + 6) = 0, we are perfectly positioned to apply the zero product property. This property, as we've discussed, states that if the product of two or more factors is zero, then at least one of the factors must be zero. In our case, the two factors are (x - 1) and (x + 6), and their product is zero. Therefore, according to the zero product property, either (x - 1) = 0 or (x + 6) = 0 (or both). This principle allows us to break down the single quadratic equation into two simpler linear equations, each of which can be solved independently to find the solutions for x. This is the power and elegance of the zero product property in action.
To find the solutions, we simply need to solve each of these linear equations. Let's start with the first equation, (x - 1) = 0. To isolate x, we add 1 to both sides of the equation. This gives us x - 1 + 1 = 0 + 1, which simplifies to x = 1. So, one solution to the original quadratic equation is x = 1. Now, let's move on to the second equation, (x + 6) = 0. To isolate x in this case, we subtract 6 from both sides of the equation. This gives us x + 6 - 6 = 0 - 6, which simplifies to x = -6. Therefore, the second solution to the original quadratic equation is x = -6.
We have now found two solutions for x: x = 1 and x = -6. These are the values of x that make the original equation, (x+2)(x+3)=12, true. To verify this, we can substitute each value back into the original equation and check if the equation holds. For x = 1, the equation becomes (1+2)(1+3)=12, which simplifies to (3)(4)=12, which is indeed true. For x = -6, the equation becomes (-6+2)(-6+3)=12, which simplifies to (-4)(-3)=12, which is also true. This verification step is a good practice to ensure that the solutions we've found are correct and that we haven't made any errors in our calculations. Thus, we can confidently conclude that the solutions to the equation (x+2)(x+3)=12 are x = 1 and x = -6. These solutions correspond to option B in the given choices.
Conclusion The Power of the Zero Product Property
In conclusion, we have successfully used the zero product property to find the solutions to the equation (x+2)(x+3)=12. This journey involved several key steps, each building upon the previous one. We started by understanding the zero product property itself, recognizing its fundamental role in solving equations where a product of factors equals zero. We then applied this property to our specific equation, which required us to first transform the equation into a suitable form. This transformation involved expanding the product, simplifying the equation, and rearranging terms to obtain a quadratic equation in the standard form. Next, we factored the quadratic expression, breaking it down into a product of two binomials. This step was crucial because it allowed us to directly apply the zero product property.
Once the equation was in factored form, we used the zero product property to set each factor equal to zero, resulting in two simpler linear equations. Solving these linear equations gave us the solutions for x, which were x = 1 and x = -6. We then verified these solutions by substituting them back into the original equation, confirming that they indeed satisfied the equation. This entire process highlights the power and elegance of the zero product property as a tool for solving algebraic equations. It allows us to break down complex problems into manageable parts, making it a cornerstone of algebraic problem-solving.
This example serves as a testament to the importance of understanding fundamental mathematical principles. The zero product property, while seemingly simple, is a powerful concept that underpins many algebraic techniques. By mastering this property and the steps involved in applying it, students can confidently tackle a wide range of equations and problems. The ability to factor quadratic expressions, combined with the zero product property, forms a formidable toolkit for solving quadratic equations, which are prevalent in various fields of mathematics and science. Therefore, a solid grasp of these concepts is essential for anyone seeking to excel in these areas. The process we've outlined here provides a clear and systematic approach to solving such equations, empowering students to approach similar problems with confidence and skill.
Final Answer: The final answer is B. x=-6 or x=1
