91Ó°ÊÓ

Quadratic equations are a staple of algebra, featuring prominently in various mathematical problems. They are typically structured as: \[ax^2 + bx + c = 0.\]Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). This formula provides a foundation for solving numerous algebraic problems.
In the provided problem, the equation \(w^2 + 2w + 1\) fits this model, where:

• \(a = 1\)
• \(b = 2\)
• \(c = 1\)

Identifying these components helps simplify and solve the trinomial effectively.

Factoring techniques are crucial in simplifying and solving polynomial expressions. Factoring a quadratic trinomial involves rewriting it as a product of two binomials.
For the given problem, the aim is to express \(w^2 + 2w + 1\) in the form \((w + m)(w + n)\), where \(m\) and \(n\) are numbers that satisfy:

• \(m * n = c\) (the constant term)
• \(m + n = b\) (the linear coefficient)

Here, we need numbers that multiply to 1 (since \(c = 1\)) and add up to 2 (since \(b = 2\)). The numbers 1 and 1 fit this requirement, leading to the factorization: \[w^2 + 2w + 1 = (w + 1)(w + 1).\]Further simplifying, we can write it as: \[(w + 1)^2.\]

Algebraic expressions consist of numbers, variables, and operations that represent a quantity. In solving problems involving them, you often need to manipulate these expressions to find solutions.
In the context of quadratic trinomials like \(w^2 + 2w + 1\), your goal is to simplify the expression by factoring. This means breaking it down into manageable components called factors. Factorizing is valuable as it can reveal roots or solutions to the equation.
Taking the expression \(w^2 + 2w + 1\), identifying the appropriate values and steps leads us to the factored form \((w + 1)^2\). This shows that the quadratic equation has a repeated root at \(w = -1\).
Understanding these core algebraic principles ensures you'll be well-equipped to handle polynomial expressions in a variety of mathematical contexts.