Solving $x^2 - 16/25 = 0$ Equation A Step-by-Step Guide

Hey guys! Today, we're diving into a super common type of math problem: solving quadratic equations. Specifically, we're going to break down the equation $x^2 - \frac{16}{25} = 0$ step-by-step. Don't worry, it's not as scary as it looks! We’ll go through each stage, making sure you understand not just how to solve it, but why each step works. So, let's get started and make math a little less mysterious.

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what quadratic equations are. A quadratic equation is basically an equation that can be written in the general form of $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'x' is our variable. The key thing here is the $x^2$ term, which makes it quadratic. In our specific equation, $x^2 - \frac{16}{25} = 0$, we can see that 'a' is 1, 'b' is 0 (because there's no 'x' term), and 'c' is $-\frac{16}{25}$.

Why are quadratic equations important? Well, they pop up everywhere in math and real-world applications. From calculating the trajectory of a ball to designing bridges, quadratics are fundamental. Knowing how to solve them opens up a whole new world of problem-solving abilities. There are several methods to tackle these equations, including factoring, using the quadratic formula, and—the method we'll focus on today—the square root property. Each method has its strengths, and understanding them all can make you a true math whiz. So, stick with me, and let's unlock the secrets of this equation!

1. Isolate $x^2$: $x^2 = \frac{16}{25}$

The very first step in solving our equation, $x^2 - \frac{16}{25} = 0$, is to isolate the $x^2$ term. Think of it like this: we want to get $x^2$ all by itself on one side of the equation. To do this, we need to get rid of that pesky $-\frac{16}{25}$. The way we do that is by performing the opposite operation. Since we're subtracting $\frac{16}{25}$, we'll add $\frac{16}{25}$ to both sides of the equation. This is a fundamental principle in algebra: whatever you do to one side, you must do to the other to keep the equation balanced.

So, when we add $\frac{16}{25}$ to both sides, we get: $x^2 - \frac{16}{25} + \frac{16}{25} = 0 + \frac{16}{25}$. The $-\frac{16}{25}$ and $+\frac{16}{25}$ on the left side cancel each other out, leaving us with just $x^2$. On the right side, $0 + \frac{16}{25}$ simplifies to $\frac{16}{25}$. Therefore, our equation now looks like this: $x^2 = \frac{16}{25}$. We've successfully isolated $x^2$! This step is crucial because it sets us up perfectly for the next stage, where we'll use the square root property to finally solve for 'x'. Remember, the goal is always to simplify and isolate the variable we're trying to find.

2. Apply the Square Root Property of Equality: $\sqrt{x^2} = \pm \sqrt{\frac{16}{25}}$

Now that we've isolated $x^2$ and have the equation $x^2 = \frac{16}{25}$, it's time to unleash the square root property of equality. This property is our key to unlocking the value of 'x'. Essentially, it states that if two things are equal, then their square roots are also equal. But—and this is super important—we need to consider both the positive and negative square roots.

Why both positive and negative? Think about it: both $(4/5)^2$ and $(-4/5)^2$ equal $\frac{16}{25}$. So, when we take the square root, we need to account for both possibilities. That's why we slap a $\pm$ (plus or minus) symbol in front of the square root on the right side. Applying the square root property to our equation, we get $\sqrt{x^2} = \pm \sqrt{\frac{16}{25}}$.

On the left side, the square root of $x^2$ is simply the absolute value of x, which we can write as |x|. However, when we consider both positive and negative roots on the right side, we can directly write x. So, we have $x = \pm \sqrt{\frac{16}{25}}$. Now, we just need to simplify the square root on the right side. The square root of a fraction is the square root of the numerator divided by the square root of the denominator. So, $\sqrt{\frac{16}{25}}$ becomes $\frac{\sqrt{16}}{\sqrt{25}}$. We know that $\sqrt{16}$ is 4 and $\sqrt{25}$ is 5. Therefore, $\sqrt{\frac{16}{25}}$ simplifies to $\frac{4}{5}$. Don't forget the $\pm$ sign! This means we have two possible solutions for x: positive $\frac{4}{5}$ and negative $\frac{4}{5}$.

Solutions to the Equation

After walking through the steps, we've arrived at the solutions for the equation $x^2 - \frac{16}{25} = 0$. Remember, we isolated $x^2$, applied the square root property, and simplified. Now, let's clearly state what those solutions are. We found that $x = \pm \frac{4}{5}$. This means there are two solutions:

• $x = \frac{4}{5}$
• $x = -\frac{4}{5}$

These are the two values of 'x' that make the equation true. If you were to plug either $\frac{4}{5}$ or $-\frac{4}{5}$ back into the original equation, you would find that both satisfy the equation. So, we've successfully solved for 'x'! It's always a good idea to double-check your work by plugging the solutions back into the original equation to make sure they fit. This helps prevent errors and builds confidence in your problem-solving skills.

Why Option A is Incorrect

Now, let's address why option A, which suggests the solutions are $x = \frac{4}{25}$ and $x = -\frac{4}{25}$, is incorrect. The key mistake here lies in not correctly applying the square root property and simplifying. Remember, when we take the square root of $\frac{16}{25}$, we're looking for a number that, when multiplied by itself, equals $\frac{16}{25}$. While it's true that 4 squared is 16 and 5 squared is 25, the option mistakenly keeps the denominator as 25 instead of taking its square root, which is 5.

In simpler terms, $\frac{4}{25}$ multiplied by itself is $\frac{16}{625}$, not $\frac{16}{25}$. Similarly, $-\frac{4}{25}$ multiplied by itself also gives $\frac{16}{625}$. So, these values don't satisfy the original equation. The correct process involves recognizing that $\sqrt{\frac{16}{25}}$ simplifies to $\frac{\sqrt{16}}{\sqrt{25}}$, which is $\frac{4}{5}$. Understanding this difference is crucial for accurately solving quadratic equations using the square root property.

Final Answer

So, to wrap it all up, the solutions to the equation $x^2 - \frac{16}{25} = 0$ are:

$x = \frac{4}{5}$ and $x = -\frac{4}{5}$

Remember the key steps: isolate $x^2$, apply the square root property (considering both positive and negative roots), and simplify. You've now got another tool in your math belt for tackling quadratic equations! Keep practicing, and you'll become a pro in no time.

Final Answer: The final answer is $\boxed{x=\frac{4}{5} \text{ and } x=-\frac{4}{5}}$