Solving The Equation (7x - 9)(6x + 5) = 0 A Step-by-Step Guide
In the realm of mathematics, solving equations stands as a fundamental skill, a cornerstone upon which more advanced concepts are built. Equations, in their essence, are mathematical statements that assert the equality between two expressions. These expressions can encompass a variety of mathematical entities, including numbers, variables, and functions. The act of solving an equation entails the determination of the specific values that, when substituted for the variables, render the equation true. These values are known as the solutions or roots of the equation.
The significance of solving equations extends far beyond the confines of the classroom. It permeates various disciplines, including physics, engineering, economics, and computer science. In physics, equations are employed to model the motion of objects, the flow of energy, and the interactions between particles. Engineers rely on equations to design structures, circuits, and machines. Economists utilize equations to analyze market trends and predict economic behavior. Computer scientists leverage equations to develop algorithms and model complex systems.
Among the diverse array of equations, quadratic equations hold a prominent position. A quadratic equation is characterized by its highest power of the variable being 2. The general form of a quadratic equation is expressed as ax² + bx + c = 0, where a, b, and c represent constants, and 'x' signifies the variable. The solutions to a quadratic equation, often referred to as its roots, represent the values of 'x' that satisfy the equation.
Several techniques exist for solving quadratic equations, each possessing its own merits and applicability. These methods include factoring, completing the square, and the quadratic formula. Factoring involves expressing the quadratic expression as a product of two linear expressions. Completing the square entails manipulating the equation to create a perfect square trinomial. The quadratic formula, a universal solution, provides a direct means of calculating the roots, regardless of the equation's factorability.
In this comprehensive guide, we embark on a journey to unravel the intricacies of solving the equation (7x - 9)(6x + 5) = 0. We will delve into the underlying principles, explore the step-by-step methodology, and emerge with a profound understanding of the solution.
The Zero Product Property serves as a cornerstone in the realm of equation-solving, particularly when dealing with equations expressed as a product of factors. This fundamental property states that if the product of two or more factors equals zero, then at least one of the factors must be equal to zero. Mathematically, this can be expressed as follows:
If A * B = 0, then either A = 0 or B = 0 (or both).
The Zero Product Property provides a powerful tool for solving equations that are already factored or can be factored. When an equation is presented in factored form, we can directly apply the property by setting each factor equal to zero and solving the resulting simpler equations. This approach effectively breaks down a complex equation into a set of more manageable ones.
To illustrate the application of the Zero Product Property, consider the equation (x - 3)(x + 2) = 0. This equation is already factored, with the factors being (x - 3) and (x + 2). According to the Zero Product Property, for the product of these factors to equal zero, at least one of them must be zero. Therefore, we can set each factor equal to zero and solve:
x - 3 = 0 or x + 2 = 0
Solving these equations individually, we obtain:
x = 3 or x = -2
Thus, the solutions to the equation (x - 3)(x + 2) = 0 are x = 3 and x = -2. These values, when substituted back into the original equation, will satisfy the equality.
The Zero Product Property extends its applicability to equations involving more than two factors. For instance, in the equation (x + 1)(x - 4)(2x - 6) = 0, we have three factors. Applying the Zero Product Property, we set each factor equal to zero:
x + 1 = 0 or x - 4 = 0 or 2x - 6 = 0
Solving these equations, we find the solutions:
x = -1 or x = 4 or x = 3
The Zero Product Property proves particularly useful when dealing with quadratic equations that can be factored. By factoring the quadratic expression into two linear factors, we can then apply the Zero Product Property to find the solutions. For example, the quadratic equation x² - 5x + 6 = 0 can be factored as (x - 2)(x - 3) = 0. Applying the Zero Product Property, we obtain the solutions x = 2 and x = 3.
In summary, the Zero Product Property provides a fundamental and efficient method for solving equations expressed as a product of factors. By setting each factor equal to zero, we can break down complex equations into simpler ones, ultimately leading to the determination of the solutions.
Now, let's embark on the process of solving the equation (7x - 9)(6x + 5) = 0. This equation presents itself in a factored form, making it an ideal candidate for the application of the Zero Product Property. Our objective is to find the values of 'x' that satisfy this equation, the values that, when substituted, will make the product of the factors equal to zero.
Here's a step-by-step breakdown of the solution:
Step 1: Apply the Zero Product Property
The equation (7x - 9)(6x + 5) = 0 is expressed as the product of two factors: (7x - 9) and (6x + 5). According to the Zero Product Property, if the product of these factors equals zero, then at least one of them must be zero. Consequently, we can set each factor equal to zero:
7x - 9 = 0 or 6x + 5 = 0
This step transforms our single equation into two simpler linear equations, each of which can be solved independently.
Step 2: Solve the First Linear Equation
Let's focus on the first linear equation: 7x - 9 = 0. Our goal is to isolate 'x' on one side of the equation. To achieve this, we can perform the following operations:
Add 9 to both sides of the equation:
7x - 9 + 9 = 0 + 9
This simplifies to:
7x = 9
Divide both sides of the equation by 7:
7x / 7 = 9 / 7
This yields the solution:
x = 9/7
Therefore, the first solution to our equation is x = 9/7.
Step 3: Solve the Second Linear Equation
Now, let's turn our attention to the second linear equation: 6x + 5 = 0. We follow a similar procedure to isolate 'x':
Subtract 5 from both sides of the equation:
6x + 5 - 5 = 0 - 5
This simplifies to:
6x = -5
Divide both sides of the equation by 6:
6x / 6 = -5 / 6
This gives us the solution:
x = -5/6
Thus, the second solution to our equation is x = -5/6.
Step 4: State the Solutions
Having solved both linear equations, we have found the two values of 'x' that satisfy the original equation (7x - 9)(6x + 5) = 0. These solutions are:
x = 9/7 and x = -5/6
These solutions represent the points where the graph of the equation intersects the x-axis. They are the values that make the equation true.
In the realm of problem-solving, verification stands as a crucial step, a safeguard against errors and a confirmation of accuracy. Once we have obtained potential solutions to an equation, it is imperative to verify their validity by substituting them back into the original equation. This process ensures that the solutions indeed satisfy the equation, making the left-hand side equal to the right-hand side.
To verify our solutions, x = 9/7 and x = -5/6, we will substitute each value into the original equation (7x - 9)(6x + 5) = 0 and check if the equality holds true.
Verification of x = 9/7
Substitute x = 9/7 into the equation:
(7 * (9/7) - 9) * (6 * (9/7) + 5) = 0
Simplify the expression:
(9 - 9) * (54/7 + 5) = 0
0 * (54/7 + 5) = 0
0 = 0
The left-hand side equals the right-hand side, confirming that x = 9/7 is indeed a solution to the equation.
Verification of x = -5/6
Substitute x = -5/6 into the equation:
(7 * (-5/6) - 9) * (6 * (-5/6) + 5) = 0
Simplify the expression:
(-35/6 - 9) * (-5 + 5) = 0
(-35/6 - 9) * 0 = 0
0 = 0
Again, the left-hand side equals the right-hand side, verifying that x = -5/6 is also a solution to the equation.
The successful verification of both solutions provides us with confidence in their accuracy. We have rigorously confirmed that x = 9/7 and x = -5/6 are the valid solutions to the equation (7x - 9)(6x + 5) = 0.
In this comprehensive guide, we have embarked on a journey to solve the equation (7x - 9)(6x + 5) = 0. We have explored the fundamental principles of equation solving, delved into the significance of the Zero Product Property, and meticulously followed a step-by-step methodology to arrive at the solutions.
We began by understanding the essence of solving equations, recognizing its importance in various disciplines. We then focused on quadratic equations, acknowledging their prevalence and the diverse techniques available for solving them. The Zero Product Property emerged as a cornerstone, providing a powerful tool for solving equations expressed as a product of factors.
With the Zero Product Property in hand, we tackled the equation (7x - 9)(6x + 5) = 0. We systematically set each factor equal to zero, solved the resulting linear equations, and obtained the solutions x = 9/7 and x = -5/6. To ensure the accuracy of our solutions, we performed a rigorous verification process, substituting each value back into the original equation and confirming that the equality held true.
Through this comprehensive exploration, we have not only solved the specific equation (7x - 9)(6x + 5) = 0 but also gained valuable insights into the broader realm of equation solving. We have honed our problem-solving skills, enhanced our understanding of mathematical principles, and empowered ourselves to tackle future challenges with confidence.
Mastering equation solving is an ongoing endeavor, a journey of continuous learning and refinement. As we encounter new equations and mathematical concepts, we will build upon the foundation we have established, expanding our repertoire of problem-solving techniques and deepening our appreciation for the beauty and power of mathematics.
