91Ó°ÊÓ

Vieta's formulas are a remarkable tool in algebra that connects the coefficients of a polynomial to the sum and product of its roots. In the context of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \), these relationships can be exceedingly useful. For such equations, Vieta's formulas state:

• The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \).
• The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \).

This means if you know the roots of a quadratic equation, you can find important aspects of the equation, such as the sum and product, without actually solving the equation for its roots. This connection is not just limited to quadratic equations but extends to polynomials of higher degrees, making Vieta's formulas a versatile technique in polynomial algebra.

The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) can be quickly found using Vieta's formula. If the roots are given as \( r_1 \) and \( r_2 \), then the formula \( r_1 + r_2 = -\frac{b}{a} \) provides a direct way to find their sum.

In our example, with roots \( r_1 = \frac{1}{2}(2+\sqrt{5}) \) and \( r_2 = \frac{1}{2}(2-\sqrt{5}) \), the computation becomes:

• \( r_1 + r_2 = \frac{1}{2}(2+\sqrt{5}) + \frac{1}{2}(2-\sqrt{5}) = 2 \).

This means the sum of these roots is \( 2 \). The application of this concept simplifies the process of forming initial expressions without requiring detailed factorization, thus streamlining the solution process for complex problems.

Similarly, the product of the roots of a quadratic equation can be found using another of Vieta's formulas. For given roots \( r_1 \) and \( r_2 \), the product is \( r_1 \cdot r_2 = \frac{c}{a} \).

In the problem we considered, this computation required using the given roots:

• \( r_1 = \frac{1}{2}(2+\sqrt{5}) \) and \( r_2 = \frac{1}{2}(2-\sqrt{5}) \).

The product is calculated as follows:

• \( r_1 \cdot r_2 = \frac{1}{4}((2)^2 - (\sqrt{5})^2) = \frac{-1}{4} \).

This result of \(-\frac{1}{4}\) needs to be accounted for when forming the equation, often requiring adjustment of the coefficient \( a \) to keep integer values, as demonstrated by multiplying through by 4 to achieve an integer form in the equation.

An important aspect when dealing with quadratic equations is ensuring that the coefficients \( a \), \( b \), and \( c \) are integers. This makes the solution and interpretation easier and straightforward, especially in practical applications.

Although Vieta’s formulas give us the relationships involving \( a \), \( b \), \( c \) without concern for fractions, final equations need to be free of fractions, which involves ensuring integer coefficients.

In our case, with \( r_1 \cdot r_2 = \frac{-1}{4} \), a choice was made to multiply through by 4 to create the equation \( 4x^2 - 8x - 1 = 0 \).

This adjustment guarantees all coefficients remain integers, respecting the equation's form as initially required: \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) must be integers, with \( a > 0 \). Paying attention to these details is crucial in correctly aligning with given constraints and producing a valid solution.