Solving 6x² - 5x = 56 Using The Zero Product Property A Step-by-Step Guide

In the realm of algebra, the zero product property stands as a cornerstone for solving polynomial equations, particularly quadratic 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. This seemingly simple concept forms the basis for a powerful technique to find the solutions (also known as roots or x-intercepts) of many algebraic equations. In this comprehensive guide, we will delve deep into the application of the zero product property, using the equation $6x^2 - 5x = 56$ as our primary example. We will break down the steps involved, explore common pitfalls, and solidify your understanding with clear explanations and practical insights. Whether you are a student grappling with quadratic equations for the first time or a seasoned mathematician seeking a refresher, this guide will provide you with the knowledge and confidence to tackle these problems effectively. Our focus will be on ensuring a clear, step-by-step understanding, making the process accessible and demystifying the application of this crucial algebraic tool.

Understanding the Zero Product Property

At its core, the zero product property is remarkably intuitive. It asserts that if a multiplication of several terms equals zero, then at least one of those terms must be zero. Symbolically, if we have an equation of the form $A * B = 0$, then either $A = 0$ or $B = 0$ (or both). This principle is not just a mathematical abstraction; it reflects a fundamental aspect of arithmetic. Think of it this way: the only way to get a product of zero is if you multiply by zero. This holds true regardless of the complexity of the factors involved. They can be simple numbers, variables, or even complex algebraic expressions. The beauty of this property lies in its ability to transform a complex equation into simpler, more manageable ones. For instance, consider the equation $(x - 2)(x + 3) = 0$. Applying the zero product property, we can deduce that either $(x - 2) = 0$ or $(x + 3) = 0$. Now, we have two linear equations, which are significantly easier to solve than the original quadratic equation. This simplification is the key to unlocking solutions to a wide range of polynomial equations. In the context of quadratic equations, which often take the form $ax^2 + bx + c = 0$, the zero product property becomes especially powerful when we can factor the quadratic expression into two linear factors. This process of factoring is often the crucial first step in solving quadratic equations using this method. Understanding this fundamental principle is the bedrock upon which we will build our understanding of solving equations like $6x^2 - 5x = 56$.

Step-by-Step Solution for 6x² - 5x = 56

To effectively use the zero product property to solve the equation $6x^2 - 5x = 56$, we need to follow a structured approach. This involves several key steps, each building upon the previous one to ultimately lead us to the solutions. Let's break down the process into manageable parts:

1. Rearrange the Equation: Setting it to Zero

The first and foremost step is to ensure that the equation is set equal to zero. This is a crucial prerequisite for applying the zero product property. The given equation is $6x^2 - 5x = 56$. To set it to zero, we need to subtract 56 from both sides of the equation. This maintains the equality while bringing all terms to one side. The result is: $6x^2 - 5x - 56 = 0$. Now, the equation is in the standard quadratic form, $ax^2 + bx + c = 0$, where $a = 6$, $b = -5$, and $c = -56$. This form is essential for the next step, which involves factoring the quadratic expression. It's important to emphasize that this step is not merely a cosmetic change; it's a fundamental transformation that allows us to leverage the zero product property effectively. Without setting the equation to zero, the property cannot be applied directly. Think of it as preparing the canvas before painting – it's a necessary preparation for the artistic process that follows.

2. Factoring the Quadratic Expression: Unlocking the Product

Now that we have the equation in the form $6x^2 - 5x - 56 = 0$, the next critical step is to factor the quadratic expression. Factoring involves breaking down the quadratic expression into a product of two binomials. This is the heart of the solution process, as it directly enables the application of the zero product property. There are several techniques for factoring quadratic expressions, including trial and error, the AC method, and using the quadratic formula to find the roots and then working backwards. For the equation $6x^2 - 5x - 56 = 0$, the factoring process can be a bit challenging, but with practice and a systematic approach, it becomes manageable. We need to find two binomials of the form $(Ax + B)(Cx + D)$ such that when multiplied, they yield the original quadratic expression. In this case, the factored form is $(2x - 7)(3x + 8) = 0$. To verify this, we can multiply the binomials using the FOIL method (First, Outer, Inner, Last):

• First: $(2x)(3x) = 6x^2$
• Outer: $(2x)(8) = 16x$
• Inner: $(-7)(3x) = -21x$
• Last: $(-7)(8) = -56$

Combining these terms, we get $6x^2 + 16x - 21x - 56 = 6x^2 - 5x - 56$, which confirms our factoring is correct. Factoring is often the most challenging part of solving quadratic equations, but it's also the most rewarding. Once the expression is factored, the zero product property can be applied directly, leading us to the solutions.

3. Applying the Zero Product Property: Finding the Roots

With the quadratic expression successfully factored into $(2x - 7)(3x + 8) = 0$, we can now unleash the power of 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, we have two factors: $(2x - 7)$ and $(3x + 8)$. Applying the property, we can set each factor equal to zero, creating two separate linear equations:

1. $$2x - 7 = 0$$

2. $$3x + 8 = 0$$

This is a crucial step because it transforms a single quadratic equation into two simpler linear equations, which are much easier to solve. Each of these linear equations represents a potential solution to the original quadratic equation. By solving each equation independently, we will find the values of $x$ that make the entire product equal to zero. This is the essence of using the zero product property: breaking down a complex problem into simpler, solvable parts. It's like dismantling a machine to fix individual components, rather than trying to fix the whole machine at once. The clarity and simplicity that the zero product property brings to the solution process is what makes it such a valuable tool in algebra.

4. Solving for x: Isolating the Variable

Having applied the zero product property, we now have two linear equations to solve: $2x - 7 = 0$ and $3x + 8 = 0$. Solving for $x$ in each equation involves isolating the variable by performing algebraic operations on both sides. Let's tackle each equation separately:

Solving 2x - 7 = 0

1. Add 7 to both sides of the equation: $2x - 7 + 7 = 0 + 7$, which simplifies to $2x = 7$.
2. Divide both sides by 2: $rac{2x}{2} = rac{7}{2}$, which gives us $x = rac{7}{2}$.

Solving 3x + 8 = 0

1. Subtract 8 from both sides of the equation: $3x + 8 - 8 = 0 - 8$, which simplifies to $3x = -8$.
2. Divide both sides by 3: $rac{3x}{3} = rac{-8}{3}$, which gives us $x = -rac{8}{3}$.

Therefore, the two solutions for $x$ are $rac{7}{2}$ and $-rac{8}{3}$. These values are the roots of the quadratic equation, and they represent the points where the parabola defined by the equation intersects the x-axis. Solving for $x$ in these linear equations is a straightforward application of basic algebraic principles. Each step is designed to systematically isolate the variable, revealing its value. This process underscores the power of algebra in transforming equations into forms that directly reveal the solutions. By carefully applying these steps, we arrive at the values of $x$ that satisfy the original quadratic equation, completing the solution process.

The Solutions

After meticulously following the steps of rearranging, factoring, applying the zero product property, and solving for $x$, we have arrived at the solutions to the equation $6x^2 - 5x = 56$. The solutions we found are:

• x = rac{7}{2}

• x = -rac{8}{3}

These values represent the roots of the quadratic equation, which are the points where the graph of the equation intersects the x-axis. In the context of the original problem, these are the values of $x$ that make the equation true. It's important to understand what these solutions mean in a broader mathematical sense. They are not just abstract numbers; they are specific points that satisfy a particular relationship defined by the equation. Each solution, when substituted back into the original equation, will make the left-hand side equal to the right-hand side, confirming its validity. Furthermore, these solutions can have practical interpretations depending on the context of the problem. For instance, if the equation represents a physical phenomenon, the solutions might represent specific times, distances, or other relevant quantities. Thus, understanding the solutions is not just about finding the numbers; it's about interpreting their significance and implications. These solutions are the culmination of our efforts, representing the final answer to the problem we set out to solve.

Common Mistakes to Avoid

While the zero product property provides a powerful method for solving quadratic equations, it's crucial to be aware of common mistakes that can lead to incorrect solutions. Recognizing and avoiding these pitfalls is essential for mastering this technique. Here are some of the most frequent errors:

1. Forgetting to Set the Equation to Zero: This is perhaps the most common mistake. The zero product property only applies when the equation is set equal to zero. Attempting to apply the property when the equation is equal to a non-zero constant will lead to incorrect results. Always ensure that you rearrange the equation so that one side is zero before factoring.
2. Incorrect Factoring: Factoring quadratic expressions can be challenging, and errors in this step can derail the entire solution process. It's essential to double-check your factoring by multiplying the binomials back together to ensure they match the original quadratic expression. Practice and familiarity with different factoring techniques are key to avoiding this mistake.
3. Applying the Zero Product Property Prematurely: The zero product property can only be applied after the expression has been factored into a product of factors. Resist the temptation to set individual terms to zero before factoring, as this is mathematically incorrect.
4. Missing a Solution: Quadratic equations typically have two solutions. When solving the linear equations resulting from the zero product property, ensure that you find both solutions. Overlooking one solution is a common error, particularly when dealing with negative numbers or fractions.
5. Algebraic Errors in Solving Linear Equations: Even after correctly applying the zero product property, errors can occur when solving the resulting linear equations. Pay close attention to the signs and operations involved, and double-check your work to minimize the risk of mistakes.

By being mindful of these common pitfalls and practicing careful, systematic problem-solving, you can significantly improve your accuracy and confidence in using the zero product property.

Conclusion

In conclusion, the zero product property is an indispensable tool for solving quadratic equations. Its elegance lies in its simplicity: if the product of factors is zero, then at least one factor must be zero. This principle allows us to transform complex quadratic equations into simpler linear equations, which are easily solved. Throughout this guide, we've walked through a step-by-step solution of the equation $6x^2 - 5x = 56$, emphasizing the importance of each stage, from setting the equation to zero to factoring the quadratic expression and finally, applying the zero product property to find the solutions. We've also highlighted common mistakes to avoid, ensuring a thorough understanding of the process. Mastering the zero product property is not just about memorizing steps; it's about grasping the underlying mathematical principle and applying it with confidence. With practice and a clear understanding of the concepts discussed, you can confidently tackle a wide range of quadratic equations. The ability to solve these equations is a fundamental skill in algebra and has applications in various fields, from physics and engineering to economics and computer science. Therefore, investing time and effort in mastering this technique is a valuable endeavor. Remember, the key to success lies in understanding the "why" behind the "how," and with a solid grasp of the zero product property, you'll be well-equipped to unlock the solutions to many mathematical challenges.