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Quadratic equations are polynomial equations of the second degree. This means the highest power of the variable is 2. They are usually written in the standard form as \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, where \(a eq 0\). This is because if \(a\) were 0, the equation would be linear rather than quadratic. In the given exercise, the quadratic equation is \(3x^2 + 6x + 1 = 0\), where \(3\) is the coefficient for \(x^2\), \(6\) is the coefficient for \(x\), and \(1\) is the constant term.

Quadratic equations have important properties, such as their graph being a parabola. This graph opens upwards if \(a > 0\) and downwards if \(a < 0\). Understanding the nature and structure of quadratic equations helps in choosing the right methods to solve them.

Solving quadratic equations means finding the values of \(x\) that satisfy the equation. These values are also called the roots of the equation. There are different methods to find the roots, including factoring, using the quadratic formula, and completing the square.

In the exercise, the method used is completing the square. This technique involves rearranging and adjusting the equation so that it forms a perfect square trinomial on one side. After forming the perfect square, we then take the square root of both sides to find the values of \(x\).

Completing the square is particularly useful for equations that are not easily factorable and is a good skill to master as it provides insight into the derivation of the quadratic formula.

Algebraic techniques are crucial for manipulating and solving mathematical expressions and equations. Completing the square is one such example. It is a systematic method that involves several algebraic steps:

• First, make sure the quadratic equation is in the form where the coefficient of \(x^2\) is 1, either by dividing through by \(a\) or simply if \(a=1\) initially.
• Next, rearrange the terms to focus on the \(x^2\) and \(x\) terms, and calculate half of the coefficient of \(x\), square it, and add it to both sides to maintain equality.
• This forms a perfect square trinomial, which can be expressed as \((x - b)^2\) on one side of the equation.
• Finally, solve the equation by taking the square root of both sides and simplifying to find the possible values for \(x\).

By mastering these algebraic techniques, not only can completing the square become second nature, but solving various types of quadratic equations and understanding their properties will also become easier.