Solving X²=9/16: The First Step Explained

Hey everyone! Let's dive into solving a quadratic equation, specifically $x^2 = \frac{9}{16}$. Quadratic equations might sound intimidating, but they're really just puzzles waiting to be solved. This equation is a simple one, allowing us to focus on the fundamental first step. When we see an equation like this, we're trying to find the value(s) of x that make the equation true. So, what's the very first thing we should do?

Understanding the Initial Move

When you're faced with an equation like $x^2 = \frac{9}{16}$, your primary goal is to isolate x. That means getting x by itself on one side of the equation. To do this, you need to undo any operations that are being applied to x. In this case, x is being squared. So, how do you undo a square? You take the square root!

Why the square root? Think of it this way: the square root is the inverse operation of squaring. Just like addition undoes subtraction, and multiplication undoes division, the square root undoes the square. When you take the square root of $x^2$, you get x (or technically, the absolute value of x, but we'll get to that in a bit).

Now, here's a crucial point: whatever you do to one side of an equation, you must do to the other side to keep the equation balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. So, if we take the square root of the left side ($x^2$), we must also take the square root of the right side ($\frac{9}{16}$). This maintains the equality.

Therefore, the first and most logical step in solving the quadratic equation $x^2 = \frac{9}{16}$ is to take the square root of both sides of the equation. This immediately starts the process of isolating x and finding its possible values. It's a direct application of inverse operations to simplify the equation and bring us closer to the solution. Other options like adding or subtracting $\frac{9}{16}$ to both sides don't directly help in isolating x and would lead to a more complicated solving process. Squaring both sides would make the exponent even larger, which is the opposite of what we want.

Why Not Other Options?

Let's briefly look at why the other options aren't the best first step:

• Adding $\frac{9}{16}$ to both sides: This would give you $x^2 + \frac{9}{16} = \frac{9}{8}$, which doesn't help isolate x. You're just adding more terms to the left side.
• Subtracting $\frac{9}{16}$ from both sides: This would give you $x^2 - \frac{9}{16} = 0$. While this is a valid equation, it's not the first step we want to take. We could solve it from here, but taking the square root directly is more efficient.
• Squaring both sides: This would give you $x^4 = \frac{81}{256}$. This makes the equation more complicated, not less. We want to reduce the power of x, not increase it.

Taking the Square Root: A Detailed Look

Okay, so we know the first step is to take the square root of both sides. Let's actually do it and see what happens.

Starting with: $x^2 = \frac{9}{16}$

Take the square root of both sides: $\sqrt{x^2} = \sqrt{\frac{9}{16}}$

Now, here's a key point: The square root of a number has both a positive and a negative solution. For example, both 3 and -3, when squared, equal 9. Because $3*3 = 9$ and $(-3)*(-3) = 9$.

Therefore, $\sqrt{x^2}$ simplifies to $\pm x$ and $\sqrt{\frac{9}{16}}$ simplifies to $\pm \frac{3}{4}$ because $\sqrt{9}=3$ and $\sqrt{16}=4$.

So we have $\pm x = \pm \frac{3}{4}$. This means that x can be either $\frac{3}{4}$ or $-\frac{3}{4}$.

Therefore, the solutions to the quadratic equation $x^2 = \frac{9}{16}$ are $x = \frac{3}{4}$ and $x = -\frac{3}{4}$.

Why the Plus or Minus?

The plus or minus ($\pm$) symbol is super important when solving equations where you take the square root. It's because both a positive and a negative number, when squared, result in a positive number. For example, $3^2 = 9$ and $(-3)^2 = 9$. When we're solving for x, we need to consider both possibilities. If we only consider the positive root, we'll miss one of the solutions.

Consider the equation $x^2 = 4$. If we only took the positive square root, we'd get $x = 2$. But what about $x = -2$? Well, $(-2)^2 = 4$ as well! So, x can be either 2 or -2. That's why we need the $\pm$ symbol.

Stepping Beyond: General Tips for Quadratic Equations

While $x^2 = \frac{9}{16}$ is a relatively simple quadratic equation, the principles we've discussed apply to more complex equations as well. Here are some general tips for solving quadratic equations:

1. Isolate the Squared Term: If your equation has other terms besides the squared term (like $ax^2 + bx + c = 0$), your first goal is often to isolate the $x^2$ term, or at least get the equation into a standard form where you can apply methods like factoring or the quadratic formula.
2. Consider Factoring: If your quadratic equation is in the form $ax^2 + bx + c = 0$, try factoring it. Factoring involves breaking down the quadratic expression into a product of two binomials. If you can factor the equation, you can easily find the solutions by setting each binomial equal to zero.
3. Use the Quadratic Formula: If you can't factor the equation, the quadratic formula is your best friend. The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula will give you the solutions to any quadratic equation, regardless of whether it can be factored or not.
4. Complete the Square: Completing the square is another method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial on one side. This method is particularly useful when the coefficient of the $x^2$ term is 1.
5. Check Your Answers: Always, always check your answers by plugging them back into the original equation. This will help you catch any errors you might have made along the way.

Conclusion

So, to recap, the very first step in solving the quadratic equation $x^2 = \frac{9}{16}$ is to take the square root of both sides of the equation. Remember to consider both the positive and negative square roots to find all possible solutions. By understanding this fundamental step and the reasons behind it, you'll be well on your way to mastering quadratic equations! Keep practicing, and you'll become a quadratic equation-solving pro in no time!

Remember, solving quadratic equations is a fundamental skill in mathematics with wide-ranging applications in various fields like physics, engineering, and computer science. Mastering this skill will not only help you in your math classes but also provide you with a valuable tool for problem-solving in real-world scenarios. Keep exploring, keep learning, and keep pushing your boundaries!