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 is done by 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 you have \( (x - 2)(x + 3) = 0 \), you can easily find the roots by setting each factor to zero. - **Finding Roots**: Factoring allows you to find the roots of the quadratic equation quickly. The solutions to the equation \( ax^2 + bx + c = 0 \) can be found by setting each factor to zero. - **Graphing**: Factored form makes it easier to graph the quadratic function, as the x-intercepts (roots) can be directly identified from the factors. - **Identifying Properties**: Factoring can help in identifying the vertex and axis of symmetry of the parabola represented by the quadratic expression. 4. **Example**: Consider the quadratic expression \( x^2 - 5x + 6 \). - To factor, 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 \( x = 2 \) and \( x = 3 \). In summary, factoring a quadratic expression simplifies the process of solving, graphing, and analyzing the expression, making it a valuable tool in algebra.