Solving The Equation √12x + 24 = 6x A Step By Step Guide
Solving equations involving square roots can sometimes seem daunting, but with a systematic approach and a clear understanding of the underlying principles, it becomes a manageable task. This article will delve into the step-by-step process of solving the equation √12x + 24 = 6x, providing a detailed explanation of each stage and highlighting key considerations to ensure accuracy. We will also explore the importance of checking for extraneous solutions, which are solutions obtained during the solving process that do not satisfy the original equation. So, let's embark on this mathematical journey and unravel the intricacies of solving radical equations.
Understanding Radical Equations
Before diving into the specifics of our equation, it's essential to grasp the fundamental concept of radical equations. A radical equation is an equation in which the variable appears inside a radical expression, most commonly a square root. The challenge in solving these equations lies in isolating the variable from within the radical. To achieve this, we often employ the technique of squaring both sides of the equation, which eliminates the square root. However, this process can sometimes introduce extraneous solutions, necessitating the crucial step of verification. Therefore, a meticulous approach is paramount to ensure that the solutions obtained are indeed valid.
Step-by-Step Solution of √12x + 24 = 6x
1. Isolate the Radical
The first step in solving any radical equation is to isolate the radical term on one side of the equation. In our case, the radical term, √12x + 24, is already isolated on the left side. This simplifies our task and allows us to proceed directly to the next step.
2. Square Both Sides
To eliminate the square root, we square both sides of the equation. This is a crucial step, as it allows us to work with a simpler, non-radical equation. Squaring both sides of √12x + 24 = 6x yields:
(√12x + 24)² = (6x)²
This simplifies to:
12x + 24 = 36x²
3. Rearrange into a Quadratic Equation
Now, we have a quadratic equation. To solve it, we need to rearrange it into the standard quadratic form, which is ax² + bx + c = 0. Subtracting 12x and 24 from both sides, we get:
36x² - 12x - 24 = 0
4. Simplify the Quadratic Equation
To make the equation easier to work with, we can simplify it by dividing all terms by their greatest common divisor, which is 12. This gives us:
3x² - x - 2 = 0
5. Solve the Quadratic Equation
We now have a simplified quadratic equation that we can solve using various methods, such as factoring, completing the square, or the quadratic formula. In this case, factoring is a straightforward approach. We need to find two numbers that multiply to -6 (3 * -2) and add up to -1. These numbers are -3 and 2. So, we can factor the quadratic equation as follows:
(3x + 2)(x - 1) = 0
Setting each factor equal to zero, we get:
3x + 2 = 0 or x - 1 = 0
Solving for x, we find two potential solutions:
x = -2/3 or x = 1
6. Check for Extraneous Solutions
This is a critical step in solving radical equations. Squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. To check for extraneous solutions, we must substitute each potential solution back into the original equation and see if it holds true.
Checking x = -2/3
Substitute x = -2/3 into the original equation √12x + 24 = 6x:
√[12(-2/3) + 24] = 6(-2/3)
√(-8 + 24) = -4
√16 = -4
4 = -4
This is not true. Therefore, x = -2/3 is an extraneous solution.
Checking x = 1
Substitute x = 1 into the original equation √12x + 24 = 6x:
√[12(1) + 24] = 6(1)
√(12 + 24) = 6
√36 = 6
6 = 6
This is true. Therefore, x = 1 is a valid solution.
Conclusion
After carefully solving the equation √12x + 24 = 6x and checking for extraneous solutions, we have determined that the only valid solution is x = 1. This process highlights the importance of not only the algebraic manipulations involved in solving equations but also the crucial step of verifying solutions to ensure accuracy. Solving radical equations requires a blend of algebraic skill and careful attention to detail, particularly when dealing with the potential for extraneous solutions. By following a systematic approach, we can confidently tackle these equations and arrive at the correct solutions.
In summary, the key steps in solving a radical equation like √12x + 24 = 6x are:
- Isolate the radical on one side of the equation.
- Square both sides to eliminate the square root.
- Rearrange the equation into a standard form (quadratic, linear, etc.).
- Solve the resulting equation using appropriate methods.
- Check for extraneous solutions by substituting each potential solution back into the original equation.
By mastering these steps, you can confidently solve a wide range of radical equations and enhance your problem-solving skills in mathematics. Remember, practice makes perfect, so try solving various radical equations to solidify your understanding and build your confidence.
This comprehensive guide has provided a detailed walkthrough of solving the equation √12x + 24 = 6x, emphasizing the importance of each step and the significance of checking for extraneous solutions. By understanding the underlying principles and applying a systematic approach, you can successfully navigate the world of radical equations and expand your mathematical prowess.
