Solving Equations: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into some interesting equations and figure out how to solve them. We'll explore three different types: a quadratic equation, a square root equation, and another square root equation. Don't worry if it sounds complicated – we'll break it down into easy-to-follow steps. By the end, you'll be a pro at finding solutions!

a. Solving $x^2 = 9$: Unveiling the Secrets of Quadratics

Let's start with our first equation: $x^2 = 9$. This is a quadratic equation, which means it involves a variable raised to the power of two. Our goal is to find the value(s) of x that make this equation true. Think of it like this: what number, when multiplied by itself, equals 9?

Understanding the Basics: In mathematics, the solutions to an equation are also known as roots or zeros. These are the values that, when plugged back into the original equation, make the equation a true statement. A quadratic equation, due to the square of the variable, often has two solutions. This is because both a positive and a negative number, when squared, result in a positive value.

Step-by-Step Solution:

1. Isolate the variable: In this case, the variable ($x$) is already isolated on one side of the equation. We just need to get rid of the square.
2. Take the square root of both sides: To eliminate the square, we take the square root of both sides of the equation. Remember, when taking the square root, we need to consider both the positive and negative possibilities. This gives us: $\sqrt{x^2} = \pm\sqrt{9}$.
3. Simplify: The square root of $x^2$ is simply x, and the square root of 9 is 3. Don't forget the $\pm$ sign! So, we have $x = \pm 3$. This means x can be either 3 or -3.

The Solutions: Therefore, the solutions to the equation $x^2 = 9$ are x = 3 and x = -3. We can write this as a solution set: {3, -3}. Let's check our work:

• If x = 3, then $3^2 = 9$, which is true.
• If x = -3, then $(-3)^2 = 9$, which is also true.

So, both 3 and -3 satisfy the original equation, confirming our solutions!

This principle applies broadly in algebra: when solving equations where the variable is raised to an even power, there will often be two solutions that make the equation true, one positive and one negative.

This simple equation provides a solid foundation for understanding the principles of quadratic equations. By mastering this example, you'll be better prepared to tackle more complex quadratic problems in the future. Remember to keep practicing and exploring new concepts. The more you work with equations, the more comfortable you'll become, and your problem-solving abilities will continue to strengthen.

b. Solving $\sqrt{x} = 3$: Unveiling the Secrets of Square Roots

Now, let's explore our second equation: $\sqrt{x} = 3$. This is a square root equation, which means the variable (x) is under a square root symbol. Our goal here is to find the value of x that satisfies this equation. In other words, what number's square root is 3?

Understanding Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. However, the concept is sometimes tricky because the square root symbol only represents the principal (non-negative) square root.

Step-by-Step Solution:

1. Isolate the square root: In this case, the square root is already isolated on one side of the equation.
2. Square both sides: To eliminate the square root, we square both sides of the equation. This gives us: $(\sqrt{x})^2 = 3^2$.
3. Simplify: The square of a square root cancels out, leaving us with x. And $3^2$ is 9. Therefore, we have x = 9.

The Solution: The solution to the equation $\sqrt{x} = 3$ is x = 9. We can write this as a solution set: {9}. Let's check our work:

• If x = 9, then $\sqrt{9} = 3$, which is true.

So, 9 satisfies the original equation, confirming our solution!

This process is fundamental to solving square root equations. By isolating the square root and then squaring both sides, you can effectively eliminate the root and solve for the variable. But it's important to remember that when squaring both sides of an equation, you could introduce extraneous solutions that don't fit in the equation. That is why it's always important to check your answers when dealing with these types of equations.

Mastering square root equations, as shown here, will prove valuable when you begin handling more complex mathematical problems. Keep practicing and exploring new concepts. The more you work with equations, the more comfortable you'll become, and your problem-solving abilities will continue to strengthen.

c. Solving $\sqrt{x} = -3$: Exploring the Realm of No Solution

Finally, let's look at our third equation: $\sqrt{x} = -3$. This is another square root equation, but there's a crucial difference here. Our goal is still to find the value of x that makes the equation true, but this time, things take a different turn.

Understanding the Limitations of Square Roots: As previously discussed, the square root symbol ($\sqrt{\;}$) represents the principal (non-negative) square root. This means the result of a square root operation is always a non-negative number. For example, $\sqrt{9} = 3$, not -3. The concept is that the square root of a number can never be negative, by definition.

Step-by-Step Approach: Let's follow our usual steps and see what happens:

1. Isolate the square root: It's already isolated.
2. Square both sides: $(\sqrt{x})^2 = (-3)^2$.
3. Simplify: This simplifies to x = 9. However, when we substitute 9 back into the original equation, we get $\sqrt{9} = -3$, which simplifies to 3 = -3. This is not true.

The Conclusion: Because the square root of a number can never be negative, this equation has no solution. We represent this as an empty solution set: {}. It means there is no value for x that would make the initial equation true. The conflict arises from the fundamental rules of square roots, which restrict the output to non-negative values.

This example underscores the importance of understanding the underlying principles of mathematical operations. It also shows that not every equation will have a real number solution. Some equations may have no solution, as is the case here. This happens when the constraints of the operations contradict each other. Learning to recognize these situations helps to build a more thorough comprehension of algebraic concepts.

In this particular instance, we can conclude that the equation has no real solutions because the square root of a real number cannot be negative. The solution set is empty, indicating that there are no numbers that can fulfill the equation's requirements.

Summary of findings:

• For $x^2 = 9$, the solutions are x = 3 and x = -3.
• For $\sqrt{x} = 3$, the solution is x = 9.
• For $\sqrt{x} = -3$, there is no solution.

By following these examples, you have enhanced your problem-solving capabilities in algebra. Keep practicing different types of equations, always remember to check your work, and don’t get discouraged if you encounter equations with no solutions. Math is about the journey of learning and discovery.

If you want to review additional math concepts, here's a link to a trusted website: Khan Academy