Describe how factoring can simplify a quadratic expression.

Factoring a quadratic expression can simplify it by breaking it down into simpler components, making it easier to analyze, solve, or manipulate. Here’s how factoring works and its benefits: 1. **Definition of a Quadratic Expression**: A quadratic expression is typically in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. 2. **Factoring Process**: The goal of factoring is to express the quadratic in the form \( (px + q)(rx + s) \), where \( p \), \( q \), \( r \), and \( s \) are constants. This involves finding two numbers that multiply to \( ac \) (the product of \( a \) and \( c \)) and add to \( b \). 3. **Benefits of Factoring**: - **Simplification**: Once factored, the expression can be simplified further, especially when solving equations. For example, if \( ax^2 + bx + c = 0 \) can be factored into \( (px + q)(rx + s) = 0 \), you can set each factor to zero to find the solutions for \( x \). - **Identifying Roots**: Factoring allows you to easily identify the roots (or zeros) of the quadratic equation, which are the values of \( x \) that make the expression equal to zero. - **Graphing**: Factored form provides insight into the graph of the quadratic function, such as the x-intercepts, which are the points where the graph crosses the x-axis. 4. **Example**: Consider the quadratic expression \( x^2 - 5x + 6 \). - To factor it, we look for two numbers that multiply to \( 6 \) (the constant term) and add to \( -5 \) (the coefficient of \( x \)). The numbers \( -2 \) and \( -3 \) fit this requirement. - Thus, we can factor the expression as \( (x - 2)(x - 3) \). - This simplification allows us to easily find the roots by setting each factor to zero: \( x - 2 = 0 \) or \( x - 3 = 0 \), giving us \( x = 2 \) and \( x = 3 \). In summary, factoring a quadratic expression simplifies the process of solving equations, identifying roots, and understanding the behavior of the quadratic function.