Solving Quadratic Equations A Step-by-Step Guide

Decoding the Equation: A Step-by-Step Guide

In the realm of algebra, quadratic equations hold a prominent position. They are the mathematical expressions that paint the curves of parabolas and model countless real-world phenomena. Today, we embark on a journey to dissect and conquer the quadratic equation -7x - 60 = x² + 10x. Our mission is twofold: to rewrite this equation into its factored form and to unearth the elusive solutions that satisfy it. This exploration is not just about finding answers; it's about understanding the elegance and power of algebraic manipulation.

Preparing the Canvas: Rearranging the Equation

The first stroke in our algebraic masterpiece involves rearranging the given equation. Our goal is to bring all the terms to one side, setting the equation equal to zero. This standard form, ax² + bx + c = 0, is the key that unlocks the factoring process. To achieve this, we add 7x and 60 to both sides of the equation:

-7x - 60 + 7x + 60 = x² + 10x + 7x + 60

This simplifies to:

0 = x² + 17x + 60

Now, we have our equation in the classic quadratic form, ready for the next step.

The Art of Factoring: Unveiling the Hidden Factors

Factoring is akin to reverse multiplication – we're seeking two binomials that, when multiplied, yield our quadratic expression. The general form we're aiming for is (x + A)(x + B) = 0, where A and B are constants. The challenge lies in finding the right pair of numbers that satisfy two crucial conditions:

1. They must multiply to give the constant term (c), which in our case is 60.
2. They must add up to the coefficient of the x term (b), which is 17.

Let's embark on a quest for these magical numbers. We need to consider the factor pairs of 60:

• 1 and 60
• 2 and 30
• 3 and 20
• 4 and 15
• 5 and 12
• 6 and 10

As we scrutinize these pairs, a spotlight shines on 5 and 12. Their product is indeed 60, and their sum, a glorious 17, perfectly matches the coefficient of our x term. Thus, we have found our A and B: 5 and 12.

Therefore, we can rewrite the quadratic equation in its factored form:

(x + 5)(x + 12) = 0

This factored form is an equivalent representation of our original equation, a testament to the power of algebraic manipulation.

The Zero Product Property: Cracking the Code to Solutions

The factored form is not just a pretty transformation; it's a gateway to finding the solutions of our equation. We invoke the Zero Product Property, a cornerstone of algebra, which states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, this means that either (x + 5) = 0 or (x + 12) = 0.

Let's explore each possibility:

1. If x + 5 = 0, then subtracting 5 from both sides gives us x = -5.
2. If x + 12 = 0, then subtracting 12 from both sides gives us x = -12.

And there we have it – the solutions to our quadratic equation! x = -5 and x = -12 are the values that, when plugged back into the original equation, will make the equation true. These are the points where the parabola represented by the equation intersects the x-axis.

In Conclusion: A Triumph of Algebra

We have successfully navigated the world of quadratic equations, transforming -7x - 60 = x² + 10x into its factored form, (x + 5)(x + 12) = 0, and unearthing the solutions, x = -5 and x = -12. This journey exemplifies the beauty and utility of algebraic techniques. By rearranging, factoring, and applying the Zero Product Property, we've not only found the answers but also gained a deeper appreciation for the structure and solutions of quadratic equations. The importance of understanding quadratic equations cannot be overstated, as they form the bedrock of many mathematical and scientific models. They appear in physics, engineering, economics, and computer science, making the ability to solve them a crucial skill. Moreover, the techniques we've used today – rearranging equations, factoring quadratic expressions, and applying the zero product property – are fundamental tools in the broader landscape of algebra. Mastering these tools opens doors to tackling more complex equations and mathematical problems. Therefore, the time spent understanding and practicing these techniques is a valuable investment in one's mathematical journey. As we conclude this exploration, let's remember that mathematics is not just about finding answers; it's about developing a way of thinking, a logical and systematic approach to problem-solving. The skills we've honed today will serve us well in tackling challenges both within and beyond the realm of mathematics. The ability to factor quadratic equations and find their solutions is a powerful asset, and the confidence gained through this process will empower us to embrace future mathematical endeavors with enthusiasm and skill. Thus, let's continue to explore, question, and learn, for the world of mathematics is vast and full of wonders waiting to be discovered. And always remember, practice makes perfect, so the more we engage with these concepts, the more proficient we become. Keep exploring, keep learning, and keep unlocking the secrets of mathematics! Finally, let's not forget the elegance of the solution itself. The factored form, (x + 5)(x + 12) = 0, is a concise and insightful representation of the original equation. It reveals the roots of the equation in a clear and direct manner. This is a testament to the power of factoring – it transforms a seemingly complex expression into a simple product of binomials, making the solutions readily apparent. This ability to simplify and reveal hidden structure is a hallmark of mathematical thinking, a skill that is valuable not only in mathematics but also in various aspects of life. In conclusion, solving quadratic equations is more than just a mathematical exercise; it's a journey into the heart of algebraic thinking. It involves a blend of techniques, from rearranging and factoring to applying fundamental properties. And it's a journey that rewards us not only with solutions but also with a deeper understanding of the beauty and power of mathematics. So, let's embrace the challenge, sharpen our skills, and continue to explore the fascinating world of equations and their solutions.

Solutions of -7x - 60 = x² + 10x: x = -5 and x = -12

As we've seen, the solutions to the equation -7x - 60 = x² + 10x are x = -5 and x = -12. These are the values that make the equation true. They are also the x-intercepts of the parabola represented by the equation. Understanding how to find the solutions to equations like this is critical in mathematics.

The Significance of Solutions

The solutions to an equation are not just numbers; they are the points where the equation holds true. In the context of a quadratic equation, these solutions, also known as roots or zeros, represent the x-intercepts of the parabola defined by the equation. This geometric interpretation adds another layer of understanding to the algebraic solutions. Moreover, the solutions often have real-world significance. For instance, in physics, they might represent the time at which a projectile hits the ground. In economics, they could indicate the break-even points for a business. Therefore, the ability to find accurate solutions is not merely an academic exercise; it has practical implications in various fields.

Verifying the Solutions

To ensure the accuracy of our solutions, it's always a good practice to verify them. We can do this by substituting each solution back into the original equation and checking if the equation holds true. Let's verify x = -5:

-7(-5) - 60 = (-5)² + 10(-5)

35 - 60 = 25 - 50

-25 = -25

The equation holds true for x = -5. Now, let's verify x = -12:

-7(-12) - 60 = (-12)² + 10(-12)

84 - 60 = 144 - 120

24 = 24

The equation also holds true for x = -12. This verification process not only confirms the correctness of our solutions but also reinforces our understanding of the equation and its properties. The importance of verifying solutions cannot be overstated, as it provides a crucial check against errors and ensures the reliability of our results. It's a habit that should be cultivated in all mathematical problem-solving.

The Power of Different Solution Methods

While we solved this quadratic equation by factoring, it's important to remember that there are other methods available, such as the quadratic formula. Each method has its strengths and weaknesses, and the choice of method often depends on the specific equation and the individual's preferences. The quadratic formula, for example, is a general method that can be used to solve any quadratic equation, regardless of whether it can be easily factored. Factoring, on the other hand, can be a quicker method when the equation has integer roots and the factors are readily apparent. The ability to choose the appropriate solution method is a key skill in mathematical problem-solving. It requires an understanding of the different methods and their applicability to various types of equations. Moreover, knowing multiple methods allows for a check on the solutions – if the same answer is obtained using different methods, the confidence in the result is significantly increased. Therefore, exploring and mastering different solution methods is a valuable investment in one's mathematical toolkit.

Solutions as a Foundation for Further Exploration

The solutions to a quadratic equation are not just an end in themselves; they are a stepping stone to further exploration. They can be used to analyze the behavior of the parabola, to solve related problems, and to gain deeper insights into the underlying mathematical concepts. For instance, the solutions can be used to determine the vertex of the parabola, which is the point where the parabola reaches its maximum or minimum value. They can also be used to graph the parabola accurately. Furthermore, the techniques used to solve quadratic equations can be extended to solve other types of equations, such as polynomial equations of higher degree. Therefore, the understanding of solutions to quadratic equations is a foundational concept in mathematics. It opens doors to a wide range of applications and further studies. It's a concept that should be thoroughly understood and mastered, as it forms the basis for more advanced mathematical topics. In conclusion, the solutions x = -5 and x = -12 are not just answers; they are keys to unlocking a deeper understanding of quadratic equations and their applications. They are a testament to the power of algebraic techniques and a foundation for further mathematical exploration. So, let's continue to delve into the world of equations and their solutions, for there is much to discover and learn.

Complete the equivalent equation for $-7x - 60 = x^2 + 10x$ in factored form: (x + _)(x + _) = 0. What are the solutions of the equation $-7x - 60 = x^2 + 10x$? Express the solutions as x = _.

Solving Quadratic Equations A Step-by-Step Guide