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To solve a quadratic equation means finding the values of the variable (usually denoted as \( x \)) that make the quadratic expression equal to zero. Quadratic equations have the general 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:

• Factoring: If a quadratic equation can be factored easily, this method is quick. Find two numbers that multiply to give \( ac \) (the product of \( a \) and \( c \)) and add to give \( b \).
• Completing the Square: This involves rearranging the equation so the expression in \( x \) becomes a perfect square trinomial.
• Graphing: Sometimes, simply graphing the equation can reveal the roots as the points where the graph crosses the \( x \)-axis.
• Quadratic Formula: This method can always solve any quadratic equation and involves using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Choosing the right method often depends on the form of the quadratic equation and personal preference for simplicity or precision. In our original exercise, transforming the equation into standard form facilitated using the quadratic formula efficiently.

The discriminant is a significant part of the quadratic formula and is given by the expression \( b^2 - 4ac \) in the quadratic equation \( ax^2 + bx + c = 0 \). The discriminant helps determine the nature of the roots of a quadratic equation.
Here's how it works:

• Positive Discriminant (\( \Delta > 0 \)): Indicates that the quadratic equation has two distinct real roots.
• Zero Discriminant (\( \Delta = 0 \)): Means there is exactly one real root. The quadratic "touches" the \( x \)-axis but does not cross it.
• Negative Discriminant (\( \Delta < 0 \)): Suggests there are no real roots, but two complex (or imaginary) roots.

In our problem, the discriminant was calculated as \( \frac{129}{64} \), a positive number, indicating that the quadratic equation has two distinct real roots, which are then solved using the quadratic formula.

The quadratic formula provides a universal solution method for finding the roots of any quadratic equation \( ax^2 + bx + c = 0 \). It is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula stems from completing the square for a general quadratic equation and offers explicit solutions for \( x \) based on the coefficients \( a \), \( b \), and \( c \).

Steps to Use the Quadratic Formula:

• Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
• Calculate the discriminant \( \Delta = b^2 - 4ac \).
• Plug \( b \), \( \Delta \), and \( a \) into the quadratic formula.
• Simplify to find the values of \( x \).

In the exercise, after transforming the equation into standard quadratic form \( x^2 - \frac{65}{8}x + 1 = 0 \), the quadratic formula was used to find the roots \( x_1 = \frac{65 + \sqrt{129}}{16} \) and \( x_2 = \frac{65 - \sqrt{129}}{16} \).
This process showcases the power of the quadratic formula for solving equations efficiently, especially when other methods like factoring do not readily apply.