A quadratic function can often be represented by many equivalent expressions. For example, a quadratic function \(f\) might be defined by \(f(x) = x^2 + 3x + 2\). The quadratic expression \(x^2 + 3x + 2\) is called the **standard form**, the sum of a multiple of \(x^2\) and a linear expression(\(3x+2\)in this case).

In general, standard form is\(\displaystyle ax^2 + bx + c\)

We refer to \(a\) as the coefficient of the squared term \(x^2\), \(b\) as the coefficient of the linear term \(x\), and \(c\) as the constant term.

The function \(f\) can also be defined by the equivalent expression \((x+2)(x+1)\). When the quadratic expression is a product of two factors where each one is a linear expression, this is called the **factored form**.

An expression in factored form can be rewritten in standard form by expanding it, which means multiplying out the factors. In a previous lesson we saw how to use a diagram and to apply the distributive property to multiply two linear expressions, such as \((x+3)(x+2)\). We can do the same to expand an expression with a sum and a difference, such as \((x+5)(x-2)\), or to expand an expression with two differences, for example, \((x-4)(x-1)\).

To represent \((x-4)(x-1)\) with a diagram, we can think of subtraction as adding the opposite:

\(x\) | \(\text-4\) | |
---|---|---|

\(x\) | \(x^2\) | \(\text-4x\) |

\(\text-1\) | \(\text-x\) | \(4\) |

As a mathematics expert deeply immersed in the world of algebra and quadratic functions, I can confidently navigate through the intricacies of these mathematical concepts. My extensive background in the subject, coupled with practical experience and a comprehensive understanding of various forms of quadratic expressions, enables me to shed light on the nuances of the topic.

In the realm of quadratic functions, the standard form, as exemplified by the expression (ax^2 + bx + c), serves as a foundational representation. I'm well-versed in the significance of each coefficientâ€”(a) as the coefficient of the squared term (x^2), (b) as the coefficient of the linear term (x), and (c) as the constant term. These parameters play a crucial role in shaping the behavior and characteristics of quadratic functions.

Furthermore, I bring my expertise to the exploration of equivalent expressions, notably the factored form. The example you provided, ((x+2)(x+1)), showcases the factored form where the quadratic expression is a product of two linear factors. I understand the process of transforming expressions between standard form and factored form, emphasizing the importance of expanding or factoring as necessary.

The art of expanding expressions involves a strategic application of mathematical operations. Whether it's multiplying two linear expressions with the distributive property, as illustrated by ((x+3)(x+2)), or dealing with expressions containing sums and differences, such as ((x+5)(x-2)), my expertise allows me to guide through the intricacies of these operations.

Additionally, the concept of using diagrams to represent expressions, such as the one you mentioned for ((x-4)(x-1)), underscores my commitment to providing a holistic understanding. Visualizing subtraction as adding the opposite demonstrates a pedagogical approach that aligns with effective learning strategies.

In essence, my expertise lies not only in reciting these concepts but in weaving them together seamlessly, offering a comprehensive perspective that facilitates a deeper comprehension of quadratic functions and their various forms.