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Quadratic equations are a fundamental part of algebra, commonly expressed in the form \(ax^2 + bx + c = 0\). They are characterized by their highest degree term being a square (i.e., an \(x^2\) term). This form is essential because it describes a wide range of phenomena in mathematics and physics.
Understanding a quadratic equation involves identifying its coefficients:

• \(a\) represents the leading coefficient, attached to \(x^2\).
• \(b\) is the coefficient of the \(x\) term.
• \(c\) is the constant term.

Solutions to quadratic equations give us important information, such as roots or x-intercepts of a parabola when graphed. Factoring quads is one way to find these solutions.

Factoring techniques are methods used to break down complex expressions into simpler ones that are easier to work with. In the context of quadratic equations, factoring is particularly useful because it allows us to find the roots of the equation. Different polynomials require different approaches for factoring.
For the trinomial given in the exercise, the first step is to factor out \(-1\). This simplifies the original expression \(-r^2 + 14r - 45\) to \(r^2 - 14r + 45\).
Next, we employ a common technique for factoring trinomials, which involves finding two numbers that both add up to the middle coefficient (-14 in this case) and multiply to the constant term (45). The numbers -9 and -5 satisfy these conditions. Thus, the trinomial factors into \((r - 9)(r - 5)\).
These techniques streamline solving and understanding algebraic expressions, making them incredibly valuable for algebraic problem-solving.

Algebraic expressions are a central aspect of algebra, comprising variables, constants, and operations such as addition, subtraction, multiplication, and division. They are the building blocks of more complex mathematical statements and functions.
The expression in our exercise, \(-r^2 + 14r - 45\), is a trinomial, a specific type of polynomial consisting of three terms. Working with algebraic expressions often involves simplifying, factoring, or expanding them to solve equations and understand relationships among variables.
In this exercise, the initial expression was simplified through factoring, a common technique to transform algebraic expressions. By factoring out \(-1\) and deciphering the trinomial into \((r - 9)(r - 5)\), the expression was rearranged into a form that makes identifying solutions straightforward.
Understanding these expressions and how to manipulate them is essential for problem-solving in mathematics, facilitating deeper insights into mathematical structures and functions.