Problem 57 - Algebra A Combined Function

Question

Multiply. $$ (4 x-3)^{2} $$

Step-by-Step Solution

Answer

\((4x - 3)^2 = 16x^2 - 24x + 9.\)

Step 1: Identify the Expression Formula

Recognize that the expression \((4x - 3)^2\) is a perfect square trinomial which follows the pattern \((a - b)^2 = a^2 - 2ab + b^2.\) Here, \(a = 4x\) and \(b = 3.\)

Step 2: Apply the Formula

Using the formula \((a - b)^2 = a^2 - 2ab + b^2,\) substitute \(a = 4x\) and \(b = 3\) into the formula:\((4x)^2 - 2(4x)(3) + (3)^2.\)

Step 3: Calculate Each Term

- The first term is \((4x)^2 = 16x^2.\)- The second term is \(- 2(4x)(3) = -24x.\)- The third term is \((3)^2 = 9.\)

Step 4: Combine Terms

Combine the calculated terms to form the expanded expression:\(16x^2 - 24x + 9.\)

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that involves rewriting a polynomial as a product of its factors. This process simplifies expressions and solves equations. When we say we're 'factoring,' we're looking to express the polynomial in a simpler form, usually by breaking it down into terms multiplied together. A perfect square trinomial, such as \((a - b)^2\), is an example where factoring reveals its 'squared' form.Some helpful steps for factoring polynomials include:

• Identifying patterns like difference of squares, perfect squares, or grouping.
• Using algebraic formulas, such as \((a - b)^2 = a^2 - 2ab + b^2\).
• Checking your work by distributing (or expanding) to verify the original polynomial.

Recognizing these patterns and practicing can make factoring more intuitive and quicker. These skills are crucial when dealing with more complex algebraic equations.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, meaning the highest exponent of its variable is two. An example form is \(ax^2 + bx + c\), where \(a, b,\) and \(c\) represent constants and\(a eq 0\). Quadratics are significant in various aspects of algebra and are often encountered when solving equations by factoring.There are multiple methods for working with quadratic expressions:

• Factoring, which involves rewriting the expression as a product of binomials if possible.
• Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find solutions or roots when the expression is set to zero.
• Completing the square, a technique used to transform the expression into a perfect square trinomial.
• Graphing, which displays a parabola often needing interpretation to find its vertex and intercepts.

Understanding these concepts generically helps in various applications, such as in physics for projectile motion or in economics for modeling cost functions.

Algebraic Formulas

Algebraic formulas are key tools in simplifying and solving algebraic expressions. They provide generalized forms that can instantly transform complex expressions into more manageable parts. One of the essential algebraic formulas for the exercise we discussed is the perfect square formula:\((a - b)^2 = a^2 - 2ab + b^2\).These formulas are powerful because they:

• Reduce calculation time significantly by allowing direct substitution.
• Help identify hidden structures or patterns within expressions.
• Enable solving equations that can otherwise be complicated.

By mastering these formulas, students can efficiently approach factoring, expanding, and simplifying tasks. They serve as critical aids throughout much of higher algebra.