Describe how factoring can simplify a quadratic expression.

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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.
Factoring a quadratic expression breaks it down into simpler parts, making it easier to find its roots and understand its graph.

Describe how factoring can simplify a quadratic expression.

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