Problem 22 - Algebra A Combined Function

Question

Multiply. $$ (7 x-3)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(49x^2 - 42x + 9\).

Step 1: Understand the Formula

We need to multiply the expression \((7x - 3)^2\). This can be expanded using the formula for a square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 7x\) and \(b = 3\).

Step 2: Apply the Formula

Substitute \(a = 7x\) and \(b = 3\) into the formula: \((7x)^2 - 2(7x)(3) + (3)^2\).

Step 3: Perform Calculations

Calculate each term:- \((7x)^2 = (7)^2 \cdot x^2 = 49x^2\)- \(-2(7x)(3) = -42x\)- \((3)^2 = 9\)

Step 4: Combine the Terms

Combine all terms to get the expanded form: \(49x^2 - 42x + 9\).

Key Concepts

Binomial Expansion

Binomial expansion is a key technique in algebra that allows us to expand expressions like \((a + b)^n\) or \((a - b)^n\). It is especially useful for polynomials, which are expressions with one or more terms. When you see a binomial raised to a power, like \((7x - 3)^2\), think about the binomial expansion formula.

For the expansion of \((a - b)^2\), we use:

• \((a - b)^2 = a^2 - 2ab + b^2\).

This formula tells us how to expand the expression. It turns a simple square into something much more useful, especially when solving equations or simplifying expressions.

The key part is recognizing that in our case, \(a = 7x\) and \(b = 3\). Once we identify these values, we can plug them into the formula to simplify the expression.

Understanding binomial expansion helps in breaking down complex algebraic expressions into simpler parts, making them easier to manipulate or solve.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They can be as simple as \(x + 2\) or as complex as polynomial expressions like \((7x - 3)^2\).

Variables such as \(x\) represent numbers we do not yet know, and they can change in value. The whole idea behind algebraic expressions is to express relationships between these unknowns and the numbers.

In our exercise, we deal with an expression, \((7x - 3)^2\), which is a binomial. To simplify such expressions, we use formulas that help transform them into something more manageable. This is done by expanding the expression into parts that we can compute.

The expanded form, \(49x^2 - 42x + 9\), is still an algebraic expression, but it's easier to work with when solving equations or determining function values. Each term in this expression, like \(49x^2\), \(-42x\), and \(9\), is important as they describe different dimensions or parts of a problem.

Mathematical Formulas

In mathematics, formulas are pre-defined expressions or equations used to perform calculations and solve problems. They serve as shortcuts and provide a standard way to handle different mathematical situations.

In our example, the formula for the expansion of a square of a binomial, \((a - b)^2 = a^2 - 2ab + b^2\), helps convert \((7x - 3)^2\) into a more straightforward form. By substituting the corresponding parts, \(a = 7x\) and \(b = 3\), you can simplify the expression systematically.

Every term in the formula has significance:

• \(a^2\) represents the square of the first term.
• \(-2ab\) accounts for the cross-product of the two terms.
• \(b^2\) gives the square of the second term.

Applying these formulas correctly helps in achieving precision and efficiency when working with algebraic problems. It's vital to understand not just how to apply them, but also why they work, which builds deeper comprehension and confidence in mathematics.