Solving X^2-15x-100=0 Using The Zero Product Property

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In this comprehensive guide, we will explore how to solve quadratic equations using the zero product property. This property is a fundamental concept in algebra and provides an elegant method for finding the solutions (also known as roots) of quadratic equations that can be factored. We will focus on the specific equation x^2 - 15x - 100 = 0, breaking down each step in detail to ensure a clear understanding. This article aims to provide a step-by-step solution, enhance your problem-solving skills, and clarify the underlying principles of the zero product property. Whether you're a student tackling algebra homework or simply looking to refresh your math knowledge, this guide will equip you with the tools to confidently solve quadratic equations.

Understanding the Zero Product Property

Before diving into the specifics of our equation, let's establish a firm understanding of the zero product property. This property states that 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 follows:

If a * b* = 0, then a = 0 or b = 0 (or both).

This seemingly simple concept is the cornerstone of solving many quadratic equations. When a quadratic equation can be factored into two binomial expressions, the zero product property allows us to set each factor equal to zero and solve for the variable. This transforms a single quadratic equation into two simpler linear equations, making the solutions readily accessible. The zero product property is a powerful tool because it connects the factored form of a quadratic equation to its solutions, providing a direct pathway to finding the values of x that satisfy the equation. Its effectiveness lies in its ability to break down a complex problem into manageable parts, allowing us to isolate the variable and determine its possible values.

The Significance of Factoring

The ability to factor a quadratic equation is crucial for applying the zero product property. Factoring involves expressing a quadratic expression as the product of two binomials. For example, the quadratic expression x^2 + 5x + 6 can be factored into (x + 2)(x + 3). This factored form immediately reveals the potential solutions when the expression is set equal to zero. The process of factoring itself can sometimes be challenging, but with practice and the application of various techniques, it becomes a manageable skill. Common factoring techniques include looking for common factors, using the difference of squares pattern, and employing the trial-and-error method for more complex trinomials. Recognizing these patterns and mastering these techniques are essential for efficiently solving quadratic equations using the zero product property. The connection between factoring and the zero product property highlights the importance of developing strong factoring skills in algebra. Factoring is not just a mathematical manipulation; it is a strategic step that unlocks the solutions to quadratic equations.

Preparing the Equation

To effectively use the zero product property, it's essential to ensure that the quadratic equation is in the standard form: ax^2 + bx + c = 0. This form sets the stage for factoring, as it aligns the terms in a way that facilitates the identification of factors. If the equation is not initially in this form, algebraic manipulations such as rearranging terms or combining like terms may be necessary. Once the equation is in standard form, the next step is to carefully analyze the coefficients a, b, and c. These coefficients play a crucial role in the factoring process, guiding the selection of appropriate factors. For instance, the constant term c provides insights into the possible constant terms within the binomial factors. The coefficient b helps to determine the combination of factors that will yield the correct middle term when the binomials are multiplied. By systematically examining these coefficients, we can streamline the factoring process and increase the likelihood of finding the correct factors. Preparing the equation in this manner is a foundational step, ensuring that the zero product property can be applied accurately and efficiently.

Solving x^2 - 15x - 100 = 0

Now, let's apply the zero product property to solve the equation x^2 - 15x - 100 = 0. Our first step is to factor the quadratic expression.

Factoring the Quadratic Expression

To factor x^2 - 15x - 100, we need to find two numbers that multiply to -100 and add up to -15. This requires a bit of strategic thinking about the factors of -100. We can start by listing the pairs of factors:

  • 1 and -100
  • -1 and 100
  • 2 and -50
  • -2 and 50
  • 4 and -25
  • -4 and 25
  • 5 and -20
  • -5 and 20
  • 10 and -10

Among these pairs, the numbers -20 and 5 satisfy our conditions: (-20) * 5 = -100 and (-20) + 5 = -15. Therefore, we can factor the quadratic expression as follows:

x^2 - 15x - 100 = (x - 20)(x + 5)

The factored form of the equation is now (x - 20)(x + 5) = 0. This is a crucial step, as it sets the stage for applying the zero product property directly.

Applying the Zero Product Property

With the equation in the factored form (x - 20)(x + 5) = 0, we can now apply the zero product 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, our factors are (x - 20) and (x + 5). Therefore, we can set each factor equal to zero and solve for x:

  1. x - 20 = 0
  2. x + 5 = 0

This splits our original quadratic equation into two simple linear equations, each of which can be solved independently.

Solving the Linear Equations

Now, let's solve each of the linear equations we obtained from applying the zero product property.

  • Solving x - 20 = 0:

    To isolate x, we add 20 to both sides of the equation:

    x - 20 + 20 = 0 + 20

    x = 20

  • Solving x + 5 = 0:

    To isolate x, we subtract 5 from both sides of the equation:

    x + 5 - 5 = 0 - 5

    x = -5

Thus, we have found two solutions for x: 20 and -5. These are the values of x that make the original quadratic equation true.

Verifying the Solutions

To ensure the accuracy of our solutions, it's always a good practice to verify them by substituting them back into the original equation. This step helps to catch any potential errors made during the factoring or solving process.

  • Verifying x = 20:

    Substitute x = 20 into the original equation:

    (20)^2 - 15(20) - 100 = 0

    400 - 300 - 100 = 0

    0 = 0 (This is true)

  • Verifying x = -5:

    Substitute x = -5 into the original equation:

    (-5)^2 - 15(-5) - 100 = 0

    25 + 75 - 100 = 0

    0 = 0 (This is true)

Since both values satisfy the original equation, we can confidently conclude that our solutions are correct. This verification step underscores the importance of accuracy in mathematical problem-solving.

Conclusion

In summary, we have successfully solved the quadratic equation x^2 - 15x - 100 = 0 using the zero product property. The solutions are x = -5 and x = 20, which corresponds to option C. By understanding and applying the zero product property, we transformed a quadratic equation into two simpler linear equations, making the solutions readily accessible. This method demonstrates the power of factoring in solving quadratic equations and highlights the importance of mastering fundamental algebraic concepts. This step-by-step guide not only provides the solution to the specific equation but also equips you with a robust problem-solving strategy applicable to a wide range of quadratic equations.