Solving Equations: A Walkthrough of 3x^2 180

Equations are fundamental in mathematics, allowing us to solve for unknown variables. This article provides a detailed step-by-step guide on how to solve the equation 3x^2 180, along with additional examples and explanations to help deepen your understanding of solving quadratic equations.

Understanding the Equation: 3x^2 180

When faced with the equation 3x^2 180, our goal is to isolate the variable (x) to find its value. This particular equation is a quadratic equation, which can have at most two solutions. To solve this, we will proceed with the following steps:

Step 1: Isolating the Variable Term

First, isolate the variable term on one side of the equation by dividing both sides by the coefficient of (x^2). In this case, divide both sides by 3:

(x^2 frac{180}{3})

Carrying out the division:

(x^2 60)

Step 2: Solving for (x)

Now that we have (x^2 60), we can take the square root of both sides to solve for (x). Remember, the square root of a number can be both positive and negative:

(x pm sqrt{60})

This gives us two solutions:

To simplify the square root, note that 60 can be factored into 4 and 15:

(sqrt{60} sqrt{4 times 15} 2 sqrt{15})

Therefore:

Alternative Method: Factoring or Completing the Square

Another method involves transforming the equation into a standard quadratic form and then factoring or using the quadratic formula. Starting from the original equation:

3x^2 - 180 0

Add 180 to both sides:

3x^2 180

Now, divide both sides by 3:

x^2 60

As before, take the square root of both sides:

x plusmn; sqrt{60}

This method is a bit more complex, but it reinforces the mathematical principles involved in solving quadratic equations.

Conclusion

By following these steps, we have successfully solved the equation 3x^2 180. The solutions are x 2sqrt{15} and x -2sqrt{15}. Understanding these methods and applying them to different quadratic equations can significantly enhance your problem-solving skills in algebra.

Delve into further explorations of quadratic equations through practice and study, as consistent practice is key to mastering these mathematical concepts. For additional resources and deeper insights, consider exploring online tutorials, textbooks, or mathematics forums.