Solving Quadratic Equations Using Completing the Square Method

Quadratic equations are a fundamental part of algebra and can be solved using various methods, including completing the square. This method involves transforming the equation into a perfect square trinomial to simplify the process of finding the roots. Let's explore how to solve the given equation 3x^5/x 12 using this method.

Step-by-Step Guide to Completing the Square

To solve the equation 3x^5/x 12, we first need to simplify and rearrange it into a standard quadratic form:

Multiply the entire equation by x to eliminate the fraction: 3x^2 - 5 12x Move all the terms to one side to set the equation to zero: 3x^2 - 12x - 5 0 Divide the entire equation by 3 to make the coefficient of the first term 1: x^2 - 4x - 5/3 0 Move the constant term to the right side: x^2 - 4x 5/3 Add 2^2 to both sides to complete the square: x^2 - 4x 2^2 5/3 2^2 Factor the left side as a perfect square trinomial: (x - 2)^2 5/3 4 Simplify the right side: (x - 2)^2 17/3 Taking the square root of both sides: x - 2 ±√(17/3) Isolate the variable x: x 2 ± √(17/3)

Alternative Method

Alternatively, we can factor the quadratic equation directly:

3x^2 - 2.√3x.2√3 - 2√3^2 - 2√3^2 5 0 (√3x - 2√3)^2 - √7^2 0 This results in: (√3x - 2√3 - √7)(√3x - 2√3 √7) 0 Solving each factor: √3x - 2√3 - √7 0 leads to x 2√3 √7/√3 √3x - 2√3 √7 0 leads to x 2√3 - √7/√3

A Summary Example

Let's apply the completing the square method to the given equation 3x^2 - 12x - 5 0 for a clearer understanding:

x^2 - 4x - 5/3 0 x^2 - 4x 5/3 x^2 - 4x 4 5/3 4 (x - 2)^2 17/3 x - 2 ±√(17/3) x 2 ± √(17/3)

In conclusion, solving quadratic equations using completing the square method involves a series of steps, including rearranging the equation, completing the square, and taking the square root of both sides. This method is particularly useful for understanding the underlying algebraic principles and is widely recognized in mathematical literature.

To reinforce your understanding, practice solving similar equations and explore other algebraic methods for solving quadratic equations. By mastering the completing the square method, you'll be better equipped to tackle more complex algebraic problems.