91Ó°ÊÓ

In algebra, quadratic expressions are polynomials of degree 2. This means the highest exponent of the variable is 2. A quadratic expression typically has the form \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.
For example, in the exercise provided, we work with quadratic expressions like \( (x-4)^2 \) and \( (x+1)^2 \). Both of these can be transformed using algebraic expansion into expressions containing \( x^2 \), \( x \), and constant terms.
These components are then manipulated to simplify the expression. Understanding how to identify and manipulate quadratic expressions is crucial in simplifying more complex algebraic problems.

Algebraic expansion involves rewriting an expression in its expanded form. This process often uses distributive properties and specific formulas. For example, the formula \( (a - b)^2 = a^2 - 2ab + b^2 \) helps us expand terms like \( (x-4)^2 \).
In our exercise, we follow these steps:

• Expand \((x-4)^2\) to get \(x^2 - 8x + 16\).
• Next, expand \(6(x+1)^2\) to get \(6(x^2 + 2x + 1)\), which simplifies further to \(6x^2 + 12x + 6\).

Breaking down expressions into their simplest forms helps in combining them appropriately later. This skill is essential as it lays the foundation for solving and simplifying any polynomial equations.

Combining like terms is a fundamental technique in algebra. Like terms have the same variables raised to the same power. To combine them, simply add or subtract their coefficients.
In our exercise, after expanding, we need to subtract \( 6(x+1)^2 \) from \( (x-4)^2 \). The expanded form is: \( x^2 - 8x + 16 - (6x^2 + 12x + 6) \).
By distributing the negative sign, it becomes:
\( x^2 - 8x + 16 - 6x^2 - 12x - 6 \).
Now, let's combine like terms:

• Combine \( x^2 \)-terms: \( x^2 - 6x^2 = -5x^2 \).
• Combine \( x \)-terms: \( -8x - 12x = -20x \).
• Combine the constants: \( 16 - 6 = 10 \).

After combining, the simplified expression is: \t \( -5x^2 - 20x + 10 \).
Mastering this process will make solving algebraic expressions much more straightforward.