Solving Quadratic Equations: A Step-by-Step Guide

by Alex Johnson 50 views

Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving the equation (x + 7)(x + 17) = 0. Don't worry if this sounds intimidating; we'll break it down into easy-to-understand steps. This type of equation falls under the category of quadratic equations, albeit in a slightly disguised form. Our primary goal is to find the values of 'x' that make this equation true. These values are often referred to as the solutions, roots, or zeros of the equation. Let's get started!

Understanding the Zero Product Property

Before we jump into the solution, let's talk about the Zero Product Property. This is the key principle that unlocks this problem. The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In simpler terms, if a × b = 0, then either a = 0 or b = 0 (or both!). This property is fundamental in solving factored quadratic equations.

In our equation, (x + 7)(x + 17) = 0, we have two factors: (x + 7) and (x + 17). According to the Zero Product Property, for the product of these two factors to be zero, at least one of them must equal zero. This understanding is crucial because it allows us to split the single equation into two simpler equations, each of which we can solve independently. This strategy significantly simplifies the problem and makes it easier to find the values of 'x' that satisfy the original equation. Grasping this concept is like having a secret weapon in your algebraic toolkit, making complex problems much more manageable.

Step-by-Step Solution

Now, let's go through the steps to solve the equation (x + 7)(x + 17) = 0:

  1. Set each factor equal to zero: Based on the Zero Product Property, we set each factor equal to zero. This gives us two separate equations:

    • x + 7 = 0
    • x + 17 = 0

  2. Solve the first equation (x + 7 = 0): To isolate 'x', we subtract 7 from both sides of the equation:

    • x + 7 - 7 = 0 - 7
    • x = -7

    So, one solution to the original equation is x = -7.

  3. Solve the second equation (x + 17 = 0): Similarly, we subtract 17 from both sides to isolate 'x':

    • x + 17 - 17 = 0 - 17
    • x = -17

    Therefore, the second solution to the original equation is x = -17.

  4. The solutions: The solutions to the equation (x + 7)(x + 17) = 0 are x = -7 and x = -17.

That's it! We have successfully solved the equation. We've identified the two values of 'x' that, when substituted back into the original equation, make the equation true. Solving quadratic equations like this is a fundamental skill in algebra, providing a solid foundation for more complex mathematical concepts.

Verification of Solutions

It's always a good practice to verify your solutions. Let's substitute each solution back into the original equation to ensure they are correct.

  1. Verify x = -7: Substitute x = -7 into (x + 7)(x + 17) = 0:
    • (-7 + 7)(-7 + 17) = 0
    • (0)(10) = 0
    • 0 = 0

The equation holds true, so x = -7 is a valid solution.

  1. Verify x = -17: Substitute x = -17 into (x + 7)(x + 17) = 0:

    • (-17 + 7)(-17 + 17) = 0
    • (-10)(0) = 0
    • 0 = 0

    The equation holds true, so x = -17 is also a valid solution.

By verifying our solutions, we can be confident that we have correctly solved the equation. This step not only confirms the accuracy of our calculations but also reinforces the understanding of how the solutions satisfy the original equation. It's an excellent habit to cultivate when working with any mathematical problem, building confidence in your problem-solving abilities.

Visualizing the Solution

Understanding the solutions visually can often deepen your comprehension. Let's briefly touch upon how we can visualize the solutions to this equation. Graphically, the equation (x + 7)(x + 17) = 0 represents a parabola that intersects the x-axis at the points x = -7 and x = -17. These points are the roots or the zeros of the quadratic equation. The parabola opens upwards because the coefficient of the x² term, if we were to expand the equation, would be positive (1x² + 24x + 119 = 0). The vertex of the parabola would lie somewhere between x = -7 and x = -17, representing the minimum point of the curve.

Plotting the parabola can provide a clear picture of how the equation behaves and where its values equal zero. This graphical representation enhances the algebraic solution by providing an intuitive understanding of the function's properties. Seeing the solutions as the points where the curve crosses the x-axis reinforces the concept of roots and helps build a stronger grasp of quadratic equations.

Expanding the Equation and Understanding Standard Form

While the factored form of the equation (x + 7)(x + 17) = 0 is convenient for solving, it can also be useful to expand it into its standard form: ax² + bx + c = 0. Expanding the equation involves multiplying the factors:

(x + 7)(x + 17) = x² + 17x + 7x + 119

Combining like terms, we get:

x² + 24x + 119 = 0

This is the standard form of the quadratic equation. In this form, a = 1, b = 24, and c = 119. Knowing the standard form can be helpful when using other methods to solve quadratic equations, such as the quadratic formula or completing the square. The standard form presents the equation in a manner that highlights the coefficients, which can be useful in applying various mathematical techniques.

The ability to convert between factored and standard forms provides flexibility in solving the equation and helps you understand different aspects of its properties. Each form provides a different lens through which to view the equation, offering insights that can inform your approach to solving it. Understanding both forms contributes to a more comprehensive grasp of quadratic equations.

Alternative Solution Methods: The Quadratic Formula

While the Zero Product Property is effective for factored equations, not all quadratic equations are easily factorable. In such cases, the quadratic formula is a universal tool to find the solutions. The quadratic formula is:

x = (-b ± √(b² - 4ac)) / 2a

Where 'a', 'b', and 'c' are the coefficients from the standard form of the quadratic equation, which we found to be x² + 24x + 119 = 0. Plugging in the values:

x = (-24 ± √(24² - 4 × 1 × 119)) / (2 × 1)

x = (-24 ± √(576 - 476)) / 2

x = (-24 ± √100) / 2

x = (-24 ± 10) / 2

This gives us two solutions:

x = (-24 + 10) / 2 = -14 / 2 = -7

x = (-24 - 10) / 2 = -34 / 2 = -17

As you can see, the quadratic formula yields the same solutions as our initial method, confirming the accuracy of our results and offering an alternative approach for future problems. The Quadratic Formula can be used to solve any quadratic equation, regardless of how easily it can be factored. This tool is essential for every mathematician, ensuring that you can solve a wide range of quadratic equations efficiently.

Conclusion: Mastering Quadratic Equations

So, we've successfully solved the equation (x + 7)(x + 17) = 0! We found that the solutions are x = -7 and x = -17. By understanding the Zero Product Property and the ability to apply it, we unlocked the keys to solving this problem. We also explored expanding the equation, verifying solutions, and even touching on visual representations and alternative solution methods like the quadratic formula. Remember, practice is key! The more you work with these concepts, the more comfortable and confident you'll become.

This process is fundamental in algebra and will prepare you for more advanced math topics. Keep practicing and exploring different types of equations. Mathematics, at its core, is about problem-solving and critical thinking. The methods and techniques you learn in algebra will provide a powerful toolkit for various challenges. Keep questioning, keep practicing, and most importantly, keep enjoying the journey of learning.

For further learning, I highly recommend checking out these links for more detailed explanations and exercises: