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A quadratic equation is a type of polynomial equation that involves a variable raised to the power of two as its highest degree. Typically, a quadratic equation is expressed in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).
Quadratic equations can be solved using several methods, such as factoring, using the quadratic formula, or by completing the square, as demonstrated here. Solving quadratic equations is fundamental because they frequently appear in various mathematical contexts and real-world applications.
When faced with a problem like solving \( 3x^2 + 3x = 5 \), our goal is to find values of \( x \) that satisfy the equation. Using the method of completing the square helps to transform the equation into a form that is easier to deal with, making it possible to find the solutions systematically.

A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. For example, \( (x + a)^2 \) expands to become \( x^2 + 2ax + a^2 \), which is a perfect square trinomial.
Recognizing and creating perfect square trinomials is crucial in the process of completing the square. By carefully adjusting the equation, we transform it into a perfect square trinomial, which simplifies the solution process.
In the original exercise \( x^2 + x + \frac{1}{4} \), the expression is a perfect square trinomial. It can be rewritten as \( (x + \frac{1}{2})^2 \). The transformation involves taking half of the coefficient of \( x \) (which is 1), squaring it to get \( \frac{1}{4} \), and adding it to both sides of the equation.

The step-by-step approach to solving equations breaks down complex processes into manageable stages, especially useful for completing the square. This methodical breakdown ensures each part of the equation is manipulated correctly.
Following the steps from the exercise, the first step involved dividing the entire equation by a common factor (in this case, 3) to simplify the coefficients, resulting in \( x^2 + x = \frac{5}{3} \).
The next steps isolated the variable terms and completed the square. The strategic addition of \( \frac{1}{4} \) on both sides created the perfect square trinomial \((x + \frac{1}{2})^2\).
Finally, solving involves taking the square root of both sides and adjusting for any remaining terms to find the solutions for \( x \). Using this systematic approach provides clarity and precision in finding the solutions \( x = -\frac{1}{2} \pm \sqrt{\frac{23}{12}} \).
By breaking it down step-by-step, learners can grasp the logic and flow of the methods used, enhancing understanding and problem-solving skills.