Solving Quadratic Equations: X²/2 = X/3 Explained

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Hey math enthusiasts! Today, we're diving into a common type of equation: a quadratic equation. Specifically, we'll be tackling the equation x²/2 = x/3. Don't worry if it sounds a bit intimidating; we'll break it down step by step to make it super clear. This equation might look simple at first glance, but it's a fantastic example of how to manipulate algebraic expressions and arrive at solutions. Understanding how to solve this kind of problem is fundamental to many areas of mathematics and science, so let's get started. We'll explore the basics, the strategies, and the insights needed to solve it.

Understanding the Basics of Quadratic Equations

Before we jump into solving x²/2 = x/3, let's get on the same page about quadratic equations in general. A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we're trying to find. The key feature is the presence of the x² term, which means the highest power of the variable is 2. This is what sets it apart from linear equations (where the highest power of x is 1) and other types of equations. Quadratic equations can have zero, one, or two solutions (also called roots), depending on the values of a, b, and c. These solutions represent the values of x that make the equation true. Solving a quadratic equation involves finding these values of x. There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In our case, x²/2 = x/3, we will use a combination of algebraic manipulation and simplification.

One crucial thing to remember is that quadratic equations often have two solutions. This is because the square of both a positive and a negative number results in a positive number. For example, both 2 and -2, when squared, give you 4. Because of this, when we solve quadratic equations, we often have to account for both possibilities. This makes solving these equations a bit more involved than solving linear ones, but with practice, it becomes second nature. Additionally, understanding the nature of the solutions (real, imaginary, or repeated) gives us insights into the behavior of the equation and its corresponding graph. The solutions to a quadratic equation are the x-intercepts of the parabola that represents the equation when graphed. This gives a visual representation of the solutions and aids in the understanding of quadratic equations.

Step-by-Step Solution for x²/2 = x/3

Alright, let's get down to the business of solving x²/2 = x/3. This equation might seem a little tricky at first, but with a few simple steps, we can find the solution(s). First, to make our lives easier, we want to eliminate the fractions. To do this, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is 6. This clears the fractions and makes the equation simpler to work with. Once we've done that, we'll end up with a cleaner expression without the hassle of fractions. The next step involves rearranging the equation to fit the standard form of a quadratic equation (ax² + bx + c = 0). This means moving all terms to one side of the equation, leaving zero on the other side. This rearrangement is crucial because it allows us to use various methods to solve the equation, like factoring or the quadratic formula. Always remember that whatever you do to one side of the equation, you must do to the other to keep it balanced.

Once we've got the equation in the standard form, we can then factor, if possible. Factoring means expressing the quadratic expression as a product of two binomials. This helps us to find the roots of the equation by setting each factor equal to zero and solving for x. If factoring is not straightforward, we can use the quadratic formula, which is a foolproof way to find the solutions to any quadratic equation. The quadratic formula is a direct application and doesn't require us to factor the equation. In our specific case, the factoring might be more straightforward, so let's aim for that. The goal is to isolate 'x' and find its value(s). The solutions we find are the values of x that satisfy the original equation, meaning they make the equation true when substituted back into the original form. When you are done solving, it's always good practice to check your solution(s) to make sure they are correct. Now, let’s go through the detailed steps.

  1. Eliminate the fractions: Multiply both sides of the equation by 6.
  • 6 * (x²/2) = 6 * (x/3)
  • 3x² = 2x
  1. Rearrange into standard quadratic form: Move all terms to one side.
  • 3x² - 2x = 0
  1. Factor the equation: Factor out the common factor, which is x.
  • x(3x - 2) = 0
  1. Solve for x: Set each factor equal to zero.
  • x = 0
  • 3x - 2 = 0 → 3x = 2 → x = 2/3

Therefore, the solutions to the equation x²/2 = x/3 are x = 0 and x = 2/3.

Understanding the Solutions

Great job, we’ve solved for the solutions of the equation x²/2 = x/3! Now, let’s talk about what these solutions mean and why we have two of them. The solutions we found are x = 0 and x = 2/3. In the context of the equation, these are the values of 'x' that satisfy the equation, meaning that when you substitute them back into the original equation, the equation holds true. To verify this, let’s substitute each value into the original equation and check. For x = 0:

  • 0²/2 = 0/3
  • 0 = 0

This is true. Now, let’s verify for x = 2/3:

  • (2/3)² / 2 = (2/3) / 3
  • (4/9) / 2 = 2/9
  • 4/18 = 2/9
  • 2/9 = 2/9

This is also true. Both of these solutions work and they both satisfy the original equation, which indicates that we have the correct answers. Quadratic equations, like the one we solved, often have two solutions because of the nature of the squared term. This is why when we manipulate the equation, we need to account for both possibilities. In some cases, a quadratic equation might have only one solution (when the two roots are the same) or no real solutions (when the roots are complex). The number and nature of the solutions depend on the discriminant of the quadratic equation. Understanding the solutions provides more than just a numerical answer; it gives you the ability to understand the behavior of the equation and its implications.

Real-World Applications

Alright, let’s shift gears and look at the real world. You might be wondering,