91Ó°ÊÓ

In mathematics, having a standardized way to write equations is essential for consistency and ease of understanding. The standard form of a quadratic equation is written as \(ax^2 + bx + c = 0\), where:

• \(a\), \(b\), and \(c\) are constants,
• \(a eq 0\) to ensure the equation is indeed quadratic, implying it has terms up to the second power of \(x\).

This form allows us to easily identify the coefficients and thus make connections to other properties, such as roots and vertex. Writing a quadratic equation in this way gives us a universal language to describe parabolas, the graphs of quadratic equations.

The difference of squares is a mathematical concept widely used to simplify expressions. It refers to the identity \(a^2 - b^2 = (a - b)(a + b)\). Here:

• \(a^2\) and \(b^2\) are individual squares,
• \(a - b\) and \(a + b\) are conjugate pairs.

This property is particularly handy in simplifying quadratic equations, as it allows us to factorize certain expressions quickly. In the context of our exercise, we used this property to simplify the factors \((x - (3 - \sqrt{5}))(x - (3 + \sqrt{5}))\) into \((x - 3)^2 - (\sqrt{5})^2\). By recognizing this form, we could break down the expression into simpler terms, reaching \((x - 3)^2 - 5 = 0\). Applying this formula saves time and effort in complex algebraic manipulations.

The roots of a quadratic equation are the values of \(x\) that make the equation equal to zero. If an equation is given in the form \(ax^2 + bx + c = 0\), finding the roots helps unlock many properties of the equation's graph and its interaction with the x-axis.

• Roots can be real or complex numbers.
• They can be found using various methods, including factoring, completing the square, or using the quadratic formula.

In the problem, we begin with known roots: \(3 - \sqrt{5}\) and \(3 + \sqrt{5}\). These indicate the x-intercepts of the parabola when it is graphed. Recognizing the roots simplifies finding the equation because roots allow us to build the quadratic factored form: \((x - r_1)(x - r_2) = 0\). This leads directly to the expanded form \(x^2 - 6x + 4 = 0\), providing insights into how the quadratic relates to its graphical properties.