Solving Quadratic Equations: A Step-by-Step Guide To Solving 2x^2 + 8x - 17 = (x+2)^2

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Hey guys! Today, we're going to dive into a classic math problem: solving for x in the equation 2x^2 + 8x - 17 = (x+2)^2. Quadratic equations can seem intimidating at first, but don't worry, we'll break it down step-by-step so you can tackle any similar problem with confidence. This comprehensive guide aims to provide a clear, concise, and engaging explanation of how to solve this equation. We'll cover everything from the initial expansion and simplification to the application of the quadratic formula, ensuring you grasp each concept along the way. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what we're dealing with. We have a quadratic equation, which means it's an equation where the highest power of x is 2. Our goal is to find the value(s) of x that make the equation true. Solving quadratic equations is a fundamental skill in algebra, with applications extending to various fields such as physics, engineering, and economics. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. Our equation looks a bit different right now, but we'll soon transform it into this standard form.

Why This Equation Matters

This particular equation, 2x^2 + 8x - 17 = (x+2)^2, is a great example because it combines a few different elements we need to handle. We have an x squared term, a linear x term, a constant, and a squared binomial on the other side. This means we'll need to use multiple algebraic techniques to solve it, including expansion, simplification, and potentially the quadratic formula. Mastering such equations equips you with the skills to tackle more complex problems in mathematics and real-world applications. For example, quadratic equations are used in physics to model projectile motion and in engineering to design parabolic structures. Understanding how to solve them opens doors to more advanced studies and practical problem-solving scenarios. So, let's roll up our sleeves and see how it's done!

Step 1: Expanding the Right Side

The first thing we need to do is get rid of those parentheses on the right side of the equation. We have (x+2)^2, which means (x+2) * (x+2). To expand this, we can use the FOIL method (First, Outer, Inner, Last) or the binomial theorem. Let's use the FOIL method, which is a straightforward way to multiply two binomials.

Applying the FOIL Method

  • First: Multiply the first terms in each binomial: x * x = x^2
  • Outer: Multiply the outer terms: x * 2 = 2x
  • Inner: Multiply the inner terms: 2 * x = 2x
  • Last: Multiply the last terms: 2 * 2 = 4

Now, let's add these terms together: x^2 + 2x + 2x + 4. Simplifying this gives us x^2 + 4x + 4. So, the right side of our equation, (x+2)^2, expands to x^2 + 4x + 4. This expansion is a critical step because it allows us to combine like terms and simplify the equation into a manageable form. Without expanding, it would be difficult to isolate x and find its value. Mastering the FOIL method is essential for any algebra student, as it's frequently used in various algebraic manipulations and equation solving scenarios. It's also a fundamental building block for more advanced mathematical concepts. Thus, understanding and correctly applying the FOIL method is a key skill in your mathematical toolkit.

Step 2: Rewriting the Equation

Now that we've expanded the right side, let's rewrite our original equation: 2x^2 + 8x - 17 = x^2 + 4x + 4. Our next goal is to get all the terms on one side of the equation, so we have zero on the other side. This will put the equation in the standard quadratic form, which is essential for applying methods like the quadratic formula or factoring.

Moving Terms to One Side

To do this, we'll subtract x^2, 4x, and 4 from both sides of the equation. This keeps the equation balanced and helps us consolidate the terms. Subtracting x^2 from both sides gives us: 2x^2 - x^2 + 8x - 17 = x^2 - x^2 + 4x + 4, which simplifies to x^2 + 8x - 17 = 4x + 4. Next, we subtract 4x from both sides: x^2 + 8x - 4x - 17 = 4x - 4x + 4, simplifying to x^2 + 4x - 17 = 4. Finally, subtract 4 from both sides: x^2 + 4x - 17 - 4 = 4 - 4, resulting in our simplified quadratic equation: x^2 + 4x - 21 = 0. This transformation is a crucial step because it brings the equation into a standard form that we can work with. Recognizing and executing these kinds of algebraic manipulations are fundamental skills for solving not just quadratic equations but a wide range of mathematical problems. The ability to rearrange and simplify equations is a cornerstone of algebraic proficiency.

Step 3: Factoring the Quadratic

Now that we have the equation in the standard form x^2 + 4x - 21 = 0, we can try to factor it. Factoring involves finding two binomials that, when multiplied together, give us our quadratic expression. This method is often the quickest way to solve a quadratic equation if the expression is factorable.

Finding the Factors

We need to find two numbers that multiply to -21 (the constant term) and add up to 4 (the coefficient of the x term). Let's think about the factors of -21: 1 and -21, -1 and 21, 3 and -7, -3 and 7. The pair that adds up to 4 is -3 and 7. So, we can rewrite our quadratic equation as (x - 3)(x + 7) = 0. Factoring the quadratic is a skill that relies on pattern recognition and an understanding of how binomials multiply. It's not always straightforward, but with practice, you can become quite adept at it. This step is particularly efficient because it allows us to bypass the quadratic formula in cases where factoring is possible, saving time and effort. Being able to factor effectively also enhances your understanding of polynomial relationships and can be applied in various algebraic contexts.

Step 4: Solving for x

We've successfully factored our equation into (x - 3)(x + 7) = 0. Now, we can use the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This is a fundamental principle in algebra and is key to solving factored equations.

Applying the Zero Product Property

So, we set each factor equal to zero: x - 3 = 0 and x + 7 = 0. Solving these two simple equations gives us our solutions for x. For x - 3 = 0, we add 3 to both sides, giving us x = 3. For x + 7 = 0, we subtract 7 from both sides, giving us x = -7. Therefore, the solutions to our original equation are x = 3 and x = -7. The Zero Product Property is a powerful tool because it transforms the problem of solving a quadratic equation into solving two linear equations, which are much simpler. This principle underscores the importance of factoring, as it makes the solutions readily accessible. Understanding and applying the Zero Product Property is crucial for solving various algebraic equations and is a cornerstone concept in algebra.

Step 5: Verify the Solution

It's always a good idea to check our solutions to make sure they're correct. We can do this by plugging each value of x back into the original equation and seeing if it holds true. This verification step is crucial to avoid errors and ensures that our solutions are valid.

Checking x = 3

Let's start with x = 3. Plugging it into the original equation 2x^2 + 8x - 17 = (x+2)^2, we get: 2(3)^2 + 8(3) - 17 = (3+2)^2. Simplifying the left side: 2(9) + 24 - 17 = 18 + 24 - 17 = 25. Simplifying the right side: (3+2)^2 = (5)^2 = 25. Since both sides are equal (25 = 25), x = 3 is indeed a valid solution.

Checking x = -7

Now, let's check x = -7. Plugging it into the original equation, we get: 2(-7)^2 + 8(-7) - 17 = (-7+2)^2. Simplifying the left side: 2(49) - 56 - 17 = 98 - 56 - 17 = 25. Simplifying the right side: (-7+2)^2 = (-5)^2 = 25. Again, both sides are equal (25 = 25), so x = -7 is also a valid solution. By verifying our solutions, we've confirmed that we've solved the equation correctly. This step is not just a formality but an integral part of the problem-solving process. It instills confidence in your answer and reinforces the understanding of how the solutions satisfy the original equation. Always remember to verify your solutions whenever possible, especially in exams and critical applications.

Conclusion

So, there you have it! We've successfully solved the quadratic equation 2x^2 + 8x - 17 = (x+2)^2. We walked through each step, from expanding the equation to factoring and applying the Zero Product Property. We also took the extra step of verifying our solutions, which is always a smart move.

Key Takeaways

Solving quadratic equations can seem challenging at first, but by breaking them down into manageable steps, you can tackle even the trickiest problems. Remember, the key steps are:

  1. Expanding: Use the FOIL method to expand any squared binomials.
  2. Rewriting: Move all terms to one side to get the equation in standard form.
  3. Factoring: Try to factor the quadratic expression.
  4. Solving: Apply the Zero Product Property to find the values of x.
  5. Verifying: Plug your solutions back into the original equation to check your work.

By mastering these steps, you'll be well-equipped to solve a wide range of quadratic equations. Quadratic equations are a fundamental topic in algebra, and understanding them will serve you well in more advanced math courses and real-world applications. Keep practicing, and you'll become a quadratic equation-solving pro in no time! Remember, math is like any other skill – the more you practice, the better you get. So, keep those pencils sharp and keep solving! You've got this!