91Ó°ÊÓ

Quadratic expressions are algebraic expressions where the highest power of the variable is 2. The general form of a quadratic expression is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In quadratic expressions, the coefficient \( a \) must be non-zero, because if \( a = 0 \), the expression becomes linear rather than quadratic.
Quadratic expressions are fundamental in understanding quadratic equations. These types of expressions often appear in real-world scenarios such as calculating areas or modeling physical phenomena. They can be rewritten in different forms such as the factored form or the vertex form, each giving insight into different properties of the expression.

In our given problem, the expression \( 4x^2 - 12x + 9 \) is an example of a quadratic expression. Our goal is to understand its structure so we can manipulate it through factorization, leading us to solutions for the corresponding equation.

The roots of an equation are essentially the solutions to the equation; they satisfy the equation when substituted back in place of the variables. For quadratic equations like \( ax^2 + bx + c = 0 \), there can be two, one, or no real roots.
These roots are the points where the graph of the quadratic equation intersects the x-axis, making them critical in graphing and understanding the behavior of quadratic functions.
There are several methods to find the roots including factoring, using the quadratic formula, or completing the square.

• Factoring: Involves rewriting the expression in a product of simpler expressions.
• Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square: A method that alters the equation into a perfect square trinomial.

In our case, after factorizing the quadratic equation, we derived the roots \( x = \frac{7}{2} \) and \( x = -\frac{1}{2} \), providing us with the necessary solutions.

Factorization methods are critical in solving quadratic equations efficiently and elegantly. These methods involve expressing the quadratic expression as a product of its factors, simplifying the process of finding the roots.
There are several strategies for factorization:

• Common Factor: Look for common terms that can be factored out of the quadratic expression.
• Difference of Squares: Utilizes the identity \( a^2 - b^2 = (a-b)(a+b) \).
• Trinomials: Used when the expression is a trinomial, aiming to find two numbers that multiply to \( a \cdot c \) and add to \( b \).

In the provided exercise, our key method was to factor the expression \( 4x^2 - 12x - 7 \) to \((2x-7)(2x+1)\).
This helps us set each factor to zero, effectively splitting the quadratic equation into two simpler linear equations \(2x-7=0\) and \(2x+1=0\), which we solved independently to find the roots.