Solving The Quadratic Equation $4x^2=64$

When you encounter a quadratic equation like $4x^2 = 64$, it might seem a bit daunting at first glance. However, with a clear understanding of algebraic principles, you can break it down and find its solutions. The goal here is to isolate the variable 'x' and determine the values that make the equation true. We'll walk through the steps to solve this particular equation, making it accessible even if math isn't your strongest subject. Remember, practice is key to mastering these concepts, and by the end of this article, you'll be equipped to tackle similar problems with confidence.

Understanding Quadratic Equations

Quadratic equations are a fundamental part of algebra, and they typically take the form $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'a' is not zero. The equation $4x^2 = 64$ is a simpler form of a quadratic equation because the 'bx' term is missing (meaning b=0), and the 'c' term is also zero when we rearrange it to $4x^2 - 64 = 0$. This particular structure makes it relatively straightforward to solve. The 'x^2' term signifies that we are dealing with a quadratic, and we expect to find, at most, two solutions. These solutions represent the points where the graph of the quadratic function (a parabola) intersects the x-axis. In our case, $4x^2 = 64$ is asking for the values of 'x' that, when squared and then multiplied by 4, result in 64. It's like a puzzle where you're trying to find the missing pieces that fit the given conditions. The coefficient '4' in front of the $x^2$ term scales the parabola, and the '64' on the right side shifts its position or determines where its roots lie. Understanding these basic components helps in visualizing what we're trying to achieve mathematically. We're essentially reversing the operations performed on 'x' to find its original value(s). This process involves inverse operations like division and taking square roots.

Step-by-Step Solution

Let's dive into the process of solving $4x^2 = 64$. Our primary objective is to get 'x' by itself. The first step is to isolate the $x^2$ term. To do this, we need to undo the multiplication by 4. We achieve this by dividing both sides of the equation by 4. So, we have:

$\frac{4x^2}{4} = \frac{64}{4}$

This simplifies to:

$x^2 = 16$

Now that we have $x^2$ isolated, the next step is to find the value of 'x'. To undo the squaring of 'x', we need to take the square root of both sides of the equation. It's crucial to remember that when you take the square root of a number to solve an equation, there are two possible solutions: a positive one and a negative one. This is because both a positive number squared and its negative counterpart squared result in the same positive number. For example, $4^2 = 16$ and $(-4)^2 = 16$. Therefore, when we take the square root of 16, we must consider both possibilities:

$\sqrt{x^2} = \pm\sqrt{16}$

This leads us to:

$x = \pm 4$

So, the two solutions to the equation $4x^2 = 64$ are $x = 4$ and $x = -4$. These are the values of 'x' that, when substituted back into the original equation, make it true. We can check this: If $x=4$, then $4(4^2) = 4(16) = 64$. If $x=-4$, then $4((-4)^2) = 4(16) = 64$. Both solutions satisfy the original equation.

Analyzing the Options

Now that we've solved the equation, let's look at the provided options:

A. $x=-16$ and $x=16$ B. $x=-8$ and $x=8$ C. $x=-4$ and $x=4$ D. $x=-2$ and $x=2$

Based on our step-by-step solution, we found that the solutions are $x = 4$ and $x = -4$. Comparing this to the given options, we can see that Option C matches our findings exactly. Options A and B involve numbers that are too large, while Option D involves numbers that are too small. It's important to perform the calculations accurately to avoid selecting an incorrect option. Sometimes, multiple-choice questions are designed to test not only your ability to solve a problem but also your attention to detail and your understanding of common pitfalls, such as forgetting the negative square root.

Common Mistakes and How to Avoid Them

When solving equations, especially those involving squares and square roots, there are a few common pitfalls that students often stumble into. One of the most frequent errors is forgetting to include the negative solution when taking the square root. As we discussed, if $x^2 = k$ (where k is a positive number), then $x = \pm\sqrt{k}$. Failing to account for the negative root means you'll only find one of the two valid solutions. For instance, in our problem, if you only considered the positive square root of 16, you'd get $x=4$, missing the crucial $x=-4$. Another common mistake is performing the operations in the wrong order. For example, some might try to take the square root of $4x^2$ directly without first isolating the $x^2$ term. The square root of $4x^2$ is $2|x|$, not $2x$. It's always best to simplify the equation as much as possible and isolate the variable term before applying inverse operations like square roots. Calculation errors can also creep in, especially during the division step. Double-checking your arithmetic, such as $64 \div 4$, is essential. A quick mental check or using a calculator can prevent these simple mistakes. Finally, misinterpreting the question or the options can lead to errors. Ensure you understand what the question is asking and carefully compare your derived solution to each option. Sometimes, options might be very close, differing only by a sign or a small numerical value, making careful comparison critical. By being aware of these potential errors and consciously applying the correct procedures, you can significantly improve your accuracy when solving quadratic equations.

Conclusion

Solving the quadratic equation $4x^2 = 64$ involves a straightforward application of algebraic principles. By isolating the $x^2$ term through division and then taking the square root of both sides, we correctly identified that $x = \pm 4$. This means the two solutions are $x = 4$ and $x = -4$, which corresponds to Option C. Remember the importance of considering both positive and negative roots when dealing with square roots in equations. This fundamental concept ensures you find all possible solutions. Practice with various quadratic equations will build your confidence and proficiency.

For further exploration into quadratic equations and algebraic concepts, you can visit Khan Academy or Wolfram MathWorld.