Solving $w^2 - 7w + 12 = 0$ By The Zero-Factor Property

Introduction to the Zero-Factor Property

When dealing with quadratic equations, a powerful tool in our arsenal is the zero-factor property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In simpler terms, if we have an equation in the form a * b* = 0, then either a = 0, b = 0, or both a and b are zero. This seemingly simple concept is the cornerstone for solving many quadratic equations, particularly those that can be factored. Understanding and applying the zero-factor property efficiently is crucial for mastering algebra and higher-level mathematics. It allows us to break down complex equations into simpler, more manageable parts, ultimately leading to the solutions we seek. The significance of this property extends beyond basic algebra, playing a vital role in calculus, trigonometry, and various fields of applied mathematics and engineering.

The zero-factor property is more than just a mathematical trick; it’s a fundamental principle grounded in the structure of the real number system. The property hinges on the fact that zero is a unique number, possessing the characteristic that any number multiplied by it results in zero. This uniqueness is what allows us to deduce that at least one factor must be zero when their product is zero. The property’s application is not limited to simple algebraic equations. It appears in diverse mathematical contexts, including polynomial equations of higher degrees, systems of equations, and even in the realm of abstract algebra. Its versatility stems from its reliance on the fundamental properties of multiplication and zero, making it a universally applicable tool in mathematical problem-solving. Moreover, the zero-factor property lays the foundation for understanding more advanced concepts, such as the roots of polynomials and the behavior of functions.

The zero-factor property is an invaluable tool in mathematics due to its ability to transform complex equations into simpler forms. By factoring a quadratic equation and applying the zero-factor property, we effectively convert a single equation into two linear equations, each of which is significantly easier to solve. This transformation is a testament to the power of factorization and the elegance of the zero-factor property. The applications of the zero-factor property are not confined to academic exercises; they extend to real-world scenarios in physics, engineering, economics, and computer science. For example, in physics, it can be used to determine the equilibrium points of a system. In engineering, it can help in designing structures that can withstand specific forces. In economics, it can be used to model supply and demand curves. In computer science, it can be used to optimize algorithms. Thus, the zero-factor property is not merely an abstract mathematical concept but a practical tool with wide-ranging applications.

Factoring the Quadratic Equation $w^2 - 7w + 12 = 0$

To solve the given equation $w^2 - 7w + 12 = 0$ using the zero-factor property, the first crucial step is to factor the quadratic expression. Factoring involves rewriting the quadratic expression as a product of two binomials. In this case, we are looking for two numbers that multiply to 12 (the constant term) and add up to -7 (the coefficient of the w term). The numbers -3 and -4 satisfy these conditions, since (-3) * (-4) = 12 and (-3) + (-4) = -7. Therefore, we can factor the quadratic expression as $(w - 3)(w - 4)$. This factorization transforms the original equation into $(w - 3)(w - 4) = 0$, which is now in a form where the zero-factor property can be readily applied.

Factoring the quadratic expression $w^2 - 7w + 12$ is a critical step in solving the equation because it allows us to leverage the zero-factor property. The process of factoring involves decomposing the quadratic into two binomial factors, which, when multiplied together, yield the original quadratic expression. The key to successful factoring lies in identifying the correct pair of numbers that satisfy both the multiplication and addition criteria derived from the coefficients of the quadratic. For more complex quadratics, this might involve trial and error or the application of more systematic methods, such as the AC method or completing the square. However, for simpler quadratics like the one we are dealing with, recognizing the appropriate factors often comes down to a combination of pattern recognition and basic arithmetic skills. The factored form $(w - 3)(w - 4) = 0$ is now perfectly poised for the application of the zero-factor property, as it presents the equation as a product of two factors equal to zero.

Once the quadratic expression is successfully factored, the equation transforms into a product of two binomials set equal to zero. This transformation is significant because it allows us to apply the zero-factor property, which is the key to finding the solutions of the equation. The factored form $(w - 3)(w - 4) = 0$ indicates that the product of two expressions, $(w - 3)$ and $(w - 4)$, is zero. According to the zero-factor property, this can only be true if at least one of these expressions is equal to zero. This principle allows us to break down the original quadratic equation into two simpler linear equations, each of which can be solved independently. The ability to simplify a complex equation into more manageable parts is one of the fundamental reasons why the zero-factor property is such a valuable tool in algebra. The factored form not only facilitates the application of the zero-factor property but also provides a clearer understanding of the roots of the equation.

Applying the Zero-Factor Property

With the equation factored as $(w - 3)(w - 4) = 0$, we can now apply the zero-factor property. This property tells us that if the product of two factors is zero, then at least one of the factors must be zero. In this case, either $(w - 3) = 0$ or $(w - 4) = 0$. This gives us two separate linear equations to solve. The first equation, w - 3 = 0, can be solved by adding 3 to both sides, yielding w = 3. The second equation, w - 4 = 0, can be solved by adding 4 to both sides, resulting in w = 4. Thus, we have found two potential solutions for the original quadratic equation.

Applying the zero-factor property to the factored equation $(w - 3)(w - 4) = 0$ is the pivotal step in determining the solutions. This property serves as a bridge, connecting the factored form of the equation to its solutions. The elegance of the zero-factor property lies in its simplicity and effectiveness. It allows us to transform a single quadratic equation into two linear equations, each of which is far easier to solve. By setting each factor equal to zero, we create two independent equations that represent the potential solutions of the original equation. This transformation not only simplifies the problem but also provides a clear and direct path to the solutions. The zero-factor property is a powerful tool because it leverages the unique properties of zero in multiplication, allowing us to break down complex problems into manageable steps.

By setting each factor to zero, we generate two simple linear equations: w - 3 = 0 and w - 4 = 0. These equations are straightforward to solve using basic algebraic techniques. The first equation, w - 3 = 0, can be solved by adding 3 to both sides, isolating w and revealing the solution w = 3. Similarly, the second equation, w - 4 = 0, can be solved by adding 4 to both sides, yielding the solution w = 4. These solutions represent the values of w that make the original quadratic equation true. It's important to note that a quadratic equation can have up to two distinct real solutions, which correspond to the points where the parabola represented by the quadratic equation intersects the x-axis. In this case, we have found two distinct solutions, indicating that the parabola intersects the x-axis at two different points. The solutions w = 3 and w = 4 are the roots or zeros of the quadratic equation.

Solutions of the Equation

After applying the zero-factor property and solving the resulting linear equations, we find that the solutions to the equation $w^2 - 7w + 12 = 0$ are w = 3 and w = 4. These values of w are the roots of the quadratic equation, meaning they are the values that make the equation true. To verify these solutions, we can substitute each value back into the original equation. When w = 3, the equation becomes $3^2 - 7(3) + 12 = 9 - 21 + 12 = 0$, which is true. Similarly, when w = 4, the equation becomes $4^2 - 7(4) + 12 = 16 - 28 + 12 = 0$, which is also true. This confirms that both w = 3 and w = 4 are indeed solutions of the equation.

The solutions w = 3 and w = 4 represent the values of the variable w that satisfy the equation $w^2 - 7w + 12 = 0$. These solutions are also known as the roots or zeros of the quadratic function represented by the equation. Geometrically, these solutions correspond to the x-intercepts of the parabola defined by the quadratic function. The parabola intersects the x-axis at the points where w = 3 and w = 4. Understanding the relationship between the solutions of a quadratic equation and the graph of the corresponding quadratic function provides a deeper insight into the nature of quadratic equations. The solutions not only satisfy the algebraic equation but also have a visual representation as the points where the parabola crosses the horizontal axis.

Verifying the solutions by substituting them back into the original equation is a crucial step in the problem-solving process. This step ensures that the solutions we have found are accurate and that no errors were made during the factoring or solving process. The process of verification involves replacing the variable w in the original equation with each solution and then simplifying the resulting expression. If the expression simplifies to zero, then the solution is correct. This process not only confirms the solutions but also reinforces the understanding of the equation and its properties. In the case of our equation, substituting w = 3 and w = 4 both resulted in the equation being true, providing a strong confirmation of the accuracy of our solutions. This verification step highlights the importance of precision and attention to detail in mathematical problem-solving.

Conclusion

In summary, we have successfully solved the quadratic equation $w^2 - 7w + 12 = 0$ using the zero-factor property. The key steps involved factoring the quadratic expression into $(w - 3)(w - 4)$, applying the zero-factor property to set each factor equal to zero, solving the resulting linear equations, and verifying the solutions. This method demonstrates the power and elegance of the zero-factor property in solving quadratic equations. The solutions we found, w = 3 and w = 4, are the values that make the original equation true and represent the roots of the quadratic equation.

Solving the quadratic equation $w^2 - 7w + 12 = 0$ using the zero-factor property exemplifies a fundamental technique in algebra. This method is not only efficient but also provides a clear and logical pathway to the solutions. The process of factoring the quadratic expression is a critical skill in algebra, and the zero-factor property allows us to leverage this skill to find the roots of the equation. The ability to solve quadratic equations is essential for various applications in mathematics, science, and engineering. Quadratic equations arise in many real-world scenarios, such as projectile motion, optimization problems, and curve fitting. Thus, mastering the techniques for solving quadratic equations, including the zero-factor property, is a valuable asset in any quantitative discipline.

The zero-factor property is a cornerstone of algebraic problem-solving, and its application extends far beyond the realm of quadratic equations. This property is a fundamental principle that underlies many algebraic techniques, including solving polynomial equations of higher degrees and analyzing the behavior of functions. The ability to factor expressions and apply the zero-factor property is a critical skill for anyone pursuing advanced studies in mathematics or related fields. Furthermore, the zero-factor property highlights the importance of understanding the properties of zero in mathematical operations. Zero plays a unique role in multiplication, and its properties are essential for many mathematical proofs and theorems. The successful solution of the equation $w^2 - 7w + 12 = 0$ serves as a testament to the power and versatility of the zero-factor property and its central role in algebraic problem-solving.