Solving The Equation (4x-7)(8x+3)=0 A Step By Step Guide

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Introduction

In the realm of mathematics, solving equations is a fundamental skill. It's the cornerstone of algebra and calculus, and it's essential for anyone looking to delve deeper into mathematical concepts. In this comprehensive guide, we'll break down the process of solving the equation $(4x-7)(8x+3)=0$, ensuring you grasp the underlying principles and can tackle similar problems with confidence. We will explore the core concepts behind solving equations, focusing on the zero-product property and its application to this specific problem. By understanding this property, you'll gain a powerful tool for solving a wide range of algebraic equations.

Our main keyword in this guide is solving equations. When dealing with equations like $(4x-7)(8x+3)=0$, we're essentially looking for the values of 'x' that make the equation true. In simpler terms, we want to find the values of 'x' that, when plugged into the equation, will result in zero. This is where the zero-product property comes into play, a concept we'll explore in detail shortly. This equation is a quadratic equation in factored form. To solve it, we use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This principle allows us to break down the equation into simpler parts and solve for 'x'.

The ability to solve equations is not just limited to the classroom. It's a skill that has practical applications in various fields, from engineering and physics to economics and computer science. Whether you're calculating the trajectory of a projectile, designing a bridge, or optimizing a financial model, the ability to solve equations is crucial. Therefore, mastering this skill will undoubtedly open doors to a wide range of opportunities and deepen your understanding of the world around you. In this guide, we'll not only provide you with the steps to solve the given equation but also equip you with the knowledge to apply this skill in different contexts.

Understanding the Zero-Product Property

The zero-product property is the cornerstone of solving equations like $(4x-7)(8x+3)=0$. This property states a simple yet powerful principle: if the product of two or more factors is zero, then at least one of the factors must be zero. Mathematically, this can be expressed as: If $a imes b = 0$, then either $a = 0$ or $b = 0$ (or both). This property is incredibly useful because it allows us to break down complex equations into simpler ones.

Consider the equation $(4x-7)(8x+3)=0$. Here, we have two factors: $(4x-7)$ and $(8x+3)$. The equation tells us that the product of these two factors is zero. According to the zero-product property, this means that either $(4x-7)$ must be equal to zero, or $(8x+3)$ must be equal to zero, or both. This insight transforms our single equation into two separate, easier-to-solve equations. We can now focus on each factor individually, setting it equal to zero and solving for 'x'. This approach simplifies the process significantly and allows us to find all possible solutions for 'x'.

The beauty of the zero-product property lies in its simplicity and its broad applicability. It's not just limited to equations with two factors; it can be extended to equations with any number of factors. For instance, if we had an equation like $(x-1)(x+2)(x-3)=0$, we could still apply the same principle: at least one of the factors must be zero. This would give us three separate equations to solve: $x-1=0$, $x+2=0$, and $x-3=0$. The zero-product property is a fundamental concept in algebra, and understanding it is crucial for solving a wide variety of equations. It's a tool that every math student should have in their arsenal, and mastering it will undoubtedly make your journey through algebra much smoother.

Step-by-Step Solution of $(4x-7)(8x+3)=0$

Now, let's apply the zero-product property to solve the equation $(4x-7)(8x+3)=0$ step by step. This process involves breaking down the equation into simpler parts and solving each part individually. By following these steps, you'll not only find the solutions to this specific equation but also gain a solid understanding of how to approach similar problems.

Step 1: Apply the Zero-Product Property

The first step is to recognize that we have two factors, $(4x-7)$ and $(8x+3)$, whose product is zero. According to the zero-product property, this means that either $(4x-7)=0$ or $(8x+3)=0$. We can now write these as two separate equations:

• $4x - 7 = 0$
• $8x + 3 = 0$

This step is crucial because it transforms our original equation into two linear equations, which are much easier to solve. By applying the zero-product property, we've effectively broken down the problem into manageable pieces.

Step 2: Solve the First Equation ($4x - 7 = 0$)

To solve the equation $4x - 7 = 0$, we need to isolate 'x'. We can do this by adding 7 to both sides of the equation:

$4x - 7 + 7 = 0 + 7$

This simplifies to:

$4x = 7$

Next, we divide both sides by 4 to solve for 'x':

$4x / 4 = 7 / 4$

This gives us our first solution:

$x = 7/4$

This solution represents one of the values of 'x' that makes the original equation true. When $x = 7/4$, the factor $(4x-7)$ becomes zero, and thus the entire product $(4x-7)(8x+3)$ becomes zero.

Step 3: Solve the Second Equation ($8x + 3 = 0$)

Now, let's solve the second equation, $8x + 3 = 0$. Again, our goal is to isolate 'x'. We start by subtracting 3 from both sides:

$8x + 3 - 3 = 0 - 3$

This simplifies to:

$8x = -3$

Next, we divide both sides by 8:

$8x / 8 = -3 / 8$

This gives us our second solution:

$x = -3/8$

This is the second value of 'x' that makes the original equation true. When $x = -3/8$, the factor $(8x+3)$ becomes zero, making the entire product zero.

Step 4: Verify the Solutions

It's always a good practice to verify our solutions by plugging them back into the original equation. This ensures that we haven't made any mistakes along the way.

Let's check $x = 7/4$:

$(4(7/4) - 7)(8(7/4) + 3) = (7 - 7)(14 + 3) = (0)(17) = 0$

This confirms that $x = 7/4$ is indeed a solution.

Now, let's check $x = -3/8$:

$(4(-3/8) - 7)(8(-3/8) + 3) = (-3/2 - 7)(-3 + 3) = (-17/2)(0) = 0$

This confirms that $x = -3/8$ is also a solution.

Step 5: State the Solutions

We have found two solutions for the equation $(4x-7)(8x+3)=0$: $x = 7/4$ and $x = -3/8$. These are the only two values of 'x' that satisfy the equation.

Alternative Methods for Solving Quadratic Equations

While we've successfully solved the equation $(4x-7)(8x+3)=0$ using the zero-product property, it's important to be aware of other methods for solving quadratic equations. These methods can be particularly useful when the equation is not already in factored form. Understanding these alternative approaches will broaden your problem-solving toolkit and allow you to tackle a wider range of quadratic equations.

One common method is the quadratic formula. A quadratic equation is generally expressed in the form $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants. The quadratic formula provides a direct way to find the solutions (also called roots) of this equation. The formula is given by:

$x = (-b ± √(b^2 - 4ac)) / 2a$

This formula might look intimidating at first, but it's a powerful tool that can solve any quadratic equation, regardless of whether it can be easily factored or not. To use the quadratic formula, you simply identify the values of 'a', 'b', and 'c' from your equation and plug them into the formula. The formula will then give you the two possible values of 'x' that satisfy the equation.

Another method for solving quadratic equations is completing the square. This technique involves manipulating the equation to create a perfect square trinomial on one side. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. For example, $x^2 + 6x + 9$ is a perfect square trinomial because it can be factored into $(x + 3)^2$. Completing the square can be a bit more involved than using the quadratic formula, but it's a valuable technique to understand because it provides a deeper insight into the structure of quadratic equations.

To use this technique, you'd first rearrange the equation to have the constant term on the right side. Then, you'd add a value to both sides that makes the left side a perfect square trinomial. The value to add is half of the coefficient of the 'x' term, squared. Finally, you'd factor the perfect square trinomial, take the square root of both sides, and solve for 'x'.

While these methods are powerful, the zero-product property is often the most efficient way to solve quadratic equations that are already in factored form, as was the case with our original equation. However, understanding the quadratic formula and completing the square provides a more complete picture of how to approach quadratic equations in general.

Real-World Applications of Solving Equations

Solving equations isn't just an abstract mathematical exercise; it's a skill that has numerous real-world applications. From engineering and physics to finance and economics, the ability to solve equations is crucial for understanding and solving problems in various fields. Let's explore some specific examples of how this skill is used in practice.

In physics, equations are used to describe the motion of objects, the behavior of light and sound, and the interactions of particles. For instance, the equation for the trajectory of a projectile involves solving quadratic equations to determine the range and maximum height of the projectile. Similarly, in electrical engineering, solving equations is essential for analyzing circuits and designing electronic devices.

Consider the simple example of calculating the distance traveled by a car moving at a constant speed. The equation distance = speed × time is a fundamental equation in physics. If you know the speed and the time, you can solve for the distance. Conversely, if you know the distance and the speed, you can solve for the time. This simple example illustrates how solving equations is used to make predictions and solve problems in the real world.

In engineering, solving equations is fundamental to design and analysis. Engineers use equations to model the behavior of structures, machines, and systems. For example, civil engineers use equations to calculate the stresses and strains in bridges and buildings, ensuring their structural integrity. Mechanical engineers use equations to design engines and machines, optimizing their performance and efficiency. Chemical engineers use equations to design chemical processes and reactors, ensuring safe and efficient operation.

Finance and economics also rely heavily on solving equations. Economists use equations to model economic phenomena, such as supply and demand, inflation, and economic growth. Financial analysts use equations to value assets, manage investments, and assess risk. For example, the equation for compound interest allows you to calculate the future value of an investment based on the initial investment, the interest rate, and the time period. Similarly, equations are used to calculate loan payments, mortgage rates, and investment returns.

Computer science is another field where solving equations is essential. Computer scientists use equations to develop algorithms, model data, and solve optimization problems. For example, machine learning algorithms often involve solving complex equations to find the optimal parameters for a model. Similarly, in cryptography, equations are used to encrypt and decrypt data, ensuring secure communication.

Conclusion

In this guide, we've explored the process of solving the equation $(4x-7)(8x+3)=0$. We've seen how the zero-product property allows us to break down complex equations into simpler parts and find their solutions. We've also discussed alternative methods for solving quadratic equations, such as the quadratic formula and completing the square, and highlighted the numerous real-world applications of solving equations in various fields.

By mastering the techniques discussed in this guide, you'll not only be able to solve similar equations with confidence but also gain a deeper appreciation for the power and versatility of mathematics. Solving equations is a fundamental skill that will serve you well in your academic pursuits and beyond. Whether you're pursuing a career in science, engineering, finance, or any other field, the ability to solve equations will be a valuable asset. So, continue practicing and exploring the world of mathematics, and you'll undoubtedly discover even more exciting applications of this essential skill.