Solving X In X² + 4x - 4 = 8 A Step-by-Step Guide
Hey guys! Let's dive into a math problem today that involves solving for x in a quadratic equation. We've got the equation x² + 4x - 4 = 8, and we need to figure out which of the given options is the correct solution. Don't worry; we'll break it down step by step so it's super clear and easy to follow. So, grab your pencils and let's get started!
Understanding Quadratic Equations
Before we jump into solving the equation, let's quickly recap what quadratic equations are all about. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (x in our case) is 2. They generally have the form ax² + bx + c = 0, where a, b, and c are constants. These equations can have two, one, or no real solutions, which represent the points where the parabola (the graph of the quadratic equation) intersects the x-axis. In our equation, x² + 4x - 4 = 8, we need to manipulate it into the standard form before we can apply our solving techniques. Think of quadratic equations as puzzles – we need to rearrange the pieces to see the solution clearly. The beauty of quadratic equations lies in their applications, popping up in physics, engineering, and even finance. Understanding them opens doors to solving real-world problems, from calculating projectile motion to designing structures.
Now, why are these equations so important? Well, quadratic equations are used to model various real-world phenomena. For instance, they can describe the trajectory of a ball thrown in the air, the shape of a satellite dish, or the behavior of electrical circuits. Mastering quadratic equations isn't just about acing a math test; it's about gaining a powerful tool for understanding and interacting with the world around us. Moreover, the techniques we use to solve quadratic equations, like factoring and using the quadratic formula, are fundamental building blocks for more advanced mathematical concepts. They form the foundation for calculus, differential equations, and other higher-level math courses. So, spending the time to truly grasp quadratic equations is an investment in your future mathematical success. Plus, solving these equations can be quite satisfying – it's like cracking a code and finding the hidden solution!
Step-by-Step Solution
1. Rearrange the Equation
Our first step is to get the equation into the standard quadratic form, which is ax² + bx + c = 0. To do this, we need to move the 8 from the right side of the equation to the left side. We achieve this by subtracting 8 from both sides of the equation. Let's do it:
x² + 4x - 4 = 8 x² + 4x - 4 - 8 = 8 - 8 x² + 4x - 12 = 0
Great! Now our equation is in the standard form. We have a = 1, b = 4, and c = -12. This form is crucial because it allows us to easily apply methods like factoring or the quadratic formula.
2. Choose a Solution Method
Now that we have our equation in standard form (x² + 4x - 12 = 0), we have a couple of options for solving it: factoring or using the quadratic formula. Factoring is a great method if the quadratic expression can be easily factored into two binomials. The quadratic formula is a more general method that always works, regardless of whether the equation can be factored easily. Let's first see if we can factor our equation. We need to find two numbers that multiply to -12 and add up to 4. If we can find such numbers, factoring will be the quicker route. If not, we'll use the quadratic formula.
3. Attempt Factoring
To factor the quadratic x² + 4x - 12, we need to find two numbers that multiply to -12 (the constant term) and add up to 4 (the coefficient of the x term). Let's list the factor pairs of -12: (1, -12), (-1, 12), (2, -6), (-2, 6), (3, -4), and (-3, 4). Looking at these pairs, we can see that the pair (-2, 6) fits the bill because -2 * 6 = -12 and -2 + 6 = 4. So, we can factor the quadratic expression as (x - 2)(x + 6).
Our equation now looks like this:
(x - 2)(x + 6) = 0
4. Solve for x
To find the values of x that satisfy the equation, we set each factor equal to zero and solve for x: x - 2 = 0 and x + 6 = 0. Solving the first equation, x - 2 = 0, we add 2 to both sides to get x = 2. Solving the second equation, x + 6 = 0, we subtract 6 from both sides to get x = -6. So, our solutions are x = 2 and x = -6.
5. Verify the Solutions
It's always a good idea to check our solutions by plugging them back into the original equation. Let's verify x = 2:
(2)² + 4(2) - 4 = 8 4 + 8 - 4 = 8 8 = 8
This is true, so x = 2 is a correct solution. Now let's verify x = -6:
(-6)² + 4(-6) - 4 = 8 36 - 24 - 4 = 8 8 = 8
This is also true, so x = -6 is a correct solution as well. We've successfully solved the equation and verified our solutions!
Conclusion
Alright, guys, we did it! We successfully solved the quadratic equation x² + 4x - 4 = 8. By rearranging the equation into standard form, factoring the quadratic expression, and setting each factor to zero, we found the solutions x = -6 and x = 2. And, just to be sure, we verified these solutions by plugging them back into the original equation. So, the correct answer is A. x = -6 or x = 2. Solving quadratic equations might seem tricky at first, but with practice and a step-by-step approach, you'll become a pro in no time. Remember, the key is to break down the problem into manageable parts and use the right tools – whether it's factoring or the quadratic formula. Keep practicing, and you'll be able to tackle any quadratic equation that comes your way! Remember, math is like a muscle – the more you exercise it, the stronger it gets. So, keep those brains working and those pencils moving! You've got this!
If you ever get stuck on a quadratic equation, don't hesitate to go back through these steps or find other resources to help you out. There are tons of great videos and articles online that can provide different perspectives and explanations. And, of course, you can always ask a friend, teacher, or tutor for help. The important thing is to stay persistent and keep learning. Math is a journey, and every problem you solve is a step forward. So, celebrate your successes, learn from your mistakes, and keep on exploring the fascinating world of mathematics!
