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A quadratic equation is a second-order polynomial equation in a single variable x with a coefficient of the x-squared term being non-zero. It has the general form: \[ax^2 + bx + c = 0,\] where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The solution to a quadratic equation typically involves finding the values of x that satisfy the equation, which is often referred to as the 'roots' of the equation.

These equations are pivotal in various fields such as physics, engineering, finance, and many areas of mathematics. They are often solved using methods such as factoring, applying the quadratic formula, or completing the square, which is the method used in the example exercise. Learning to solve these equations not only aids in mathematical proficiency but also enhances problem-solving capabilities in applied sciences.

Perfect square trinomials are a specific form of quadratics that can be expressed as the square of a binomial. This means they have the form: \[ (ax+b)^2 = a^2x^2 + 2abx + b^2,\] where the middle term is twice the product of the coefficients 'a' and 'b' from the binomial 'ax+b'. Recognizing this pattern is crucial when solving quadratics by completing the square because it allows us to convert a seemingly complex quadratic equation to a simple binomial squared.

To form a perfect square trinomial from a given quadratic expression, one may need to add a specific term that allows it to be factored into this square binomial form. This is exactly what happens when we complete the square – we manipulate the equation to create this perfect square trinomial, greatly simplifying the process of finding the roots of the equation.

Solving quadratic equations is about finding the value(s) of x that make the equation true. There are several methods to solve these equations, including graphing, using the quadratic formula, factoring, and completing the square.

When the quadratic cannot be factored easily, completing the square is a reliable method that always works. It involves creating a perfect square trinomial from the original equation, as demonstrated in the textbook exercise. Once a perfect square has been formed, the equation can be solved by extracting the square root of both sides, which yields a simple linear equation that can be solved for x. This method is particularly useful when the quadratic equation does not factor neatly or when we want to derive the quadratic formula itself.

It's vital for students to understand each method's advantages and when to employ them, for efficient and accurate problem-solving. With practice, the process becomes intuitive, helping students to tackle a wide range of problems involving quadratic equations.