Solving For X: 6x^2 - 12 = 6

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Let's dive into solving the equation 6x^2 - 12 = 6. This is a quadratic equation, and we'll walk through the steps to find the value(s) of x that make this equation true. Understanding how to solve such equations is a fundamental skill in algebra, with applications ranging from basic mathematics to more complex fields like physics and engineering.

Step-by-Step Solution

First, we need to isolate the term with x^2. We can do this by adding 12 to both sides of the equation:

6x^2 - 12 + 12 = 6 + 12

This simplifies to:

6x^2 = 18

Next, we want to isolate x^2 completely. To do this, we divide both sides of the equation by 6:

(6x^2) / 6 = 18 / 6

Which simplifies to:

x^2 = 3

Now, to find x, we need to take the square root of both sides of the equation. Remember that when taking the square root, we must consider both the positive and negative roots:

√(x^2) = ±√3

This gives us:

x = ±√3

So, the solutions for x are √3 and -√3. These are the two values that, when substituted back into the original equation, will make the equation true. Understanding the concept of both positive and negative roots is crucial for solving quadratic equations accurately. Failing to consider both roots can lead to incomplete or incorrect solutions, especially in applied problems where both positive and negative values might have physical significance.

Verification

To ensure our solutions are correct, we can substitute each value back into the original equation:

For x = √3:

6(√3)^2 - 12 = 6(3) - 12 = 18 - 12 = 6

For x = -√3:

6(-√3)^2 - 12 = 6(3) - 12 = 18 - 12 = 6

Both solutions satisfy the original equation, so our solutions are correct.

Alternative Methods

While the above method is straightforward, there are alternative ways to approach this problem. One such method involves rearranging the equation into a standard quadratic form and then applying the quadratic formula, though it's generally more complex for this specific problem.

Rearranging and Factoring (If Applicable)

In some cases, a quadratic equation can be solved by factoring. However, in this case, factoring isn't the most efficient approach because the equation doesn't readily factor in a simple way. Factoring typically works best when the quadratic expression can be easily decomposed into two binomial expressions.

Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form ax^2 + bx + c = 0. The formula is:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation 6x^2 - 12 = 6, we can rewrite it as 6x^2 + 0x - 18 = 0. Here, a = 6, b = 0, and c = -18. Plugging these values into the quadratic formula:

x = (0 ± √(0^2 - 4 * 6 * -18)) / (2 * 6)

x = (± √(432)) / 12

x = (± √(144 * 3)) / 12

x = (± 12√3) / 12

x = ±√3

As you can see, the quadratic formula gives us the same solutions as our initial method, confirming the correctness of our solutions.

Common Mistakes to Avoid

When solving equations like this, there are several common mistakes to watch out for:

  1. Forgetting the Negative Root: Always remember to consider both the positive and negative square roots when solving for x. Failing to do so will result in only finding one of the two possible solutions.
  2. Incorrectly Applying the Order of Operations: Make sure to follow the correct order of operations (PEMDAS/BODMAS) when simplifying the equation. This is particularly important when dealing with exponents and square roots.
  3. Dividing Before Isolating: Avoid dividing or simplifying terms before isolating the term containing x^2. This can lead to errors in the subsequent steps.
  4. Making Arithmetic Errors: Simple arithmetic errors can easily lead to incorrect solutions. Double-check each step to ensure accuracy.

Avoiding these common mistakes will significantly improve your accuracy when solving quadratic equations.

Real-World Applications

Quadratic equations aren't just abstract mathematical concepts; they have numerous real-world applications. They are used extensively in physics to describe projectile motion, in engineering to design structures and circuits, and in economics to model supply and demand curves. Understanding how to solve quadratic equations is therefore essential for anyone pursuing a career in these fields.

For example, in physics, the trajectory of a projectile (like a ball thrown into the air) can be described by a quadratic equation. By solving this equation, you can determine the maximum height reached by the projectile, its range, and the time it takes to hit the ground. In engineering, quadratic equations are used to calculate the optimal dimensions of a bridge or the resonant frequency of an electrical circuit. These are just a few examples of how quadratic equations are used in practical applications, highlighting their importance in various scientific and technical disciplines.

Practice Problems

To solidify your understanding, here are a few practice problems:

  1. Solve for x: 4x^2 - 16 = 0
  2. Solve for x: 2x^2 - 8 = 10
  3. Solve for x: 3x^2 + 5 = 32

Working through these problems will help you develop your skills and confidence in solving quadratic equations. Remember to follow the steps outlined above and to double-check your answers to avoid common mistakes. Practice is key to mastering any mathematical concept, and quadratic equations are no exception.

Conclusion

In summary, solving the equation 6x^2 - 12 = 6 involves isolating the x^2 term, taking the square root of both sides, and considering both positive and negative roots. The solutions are x = √3 and x = -√3. By understanding the steps involved and avoiding common mistakes, you can confidently solve similar quadratic equations. The applications of these skills extend far beyond the classroom, making them invaluable in various fields.

For further learning, you can explore resources like Khan Academy's quadratic equations section to deepen your understanding: Khan Academy Quadratic Equations