Problem 43 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (2 x+3)^{2} $$

Step-by-Step Solution

Answer

The product of \((2x + 3)^2\) is \(4x^2 + 12x + 9\).

Step 1: Identify a and b

In this problem, the expression \((2x+3)^2\) is the square of the binomial \(2x+3\). So, \(a = 2x\) and \(b = 3\).

Step 2: Apply the formula for a perfect square

Substitute \(a = 2x\) and \(b = 3\) into the formula \(a^2 + 2ab + b^2\). This gives \((2x)^2 + 2*(2x)*(3) + (3)^2\).

Step 3: Simplify

Calculate \(a^2\), \(2ab\), and \(b^2\) to get \(4x^2 + 12x + 9\)

Key Concepts

Perfect Square Trinomials

Understanding perfect square trinomials is essential when working with binomial products. A perfect square trinomial is the result of squaring a binomial. It takes the form of \(a^2 + 2ab + b^2\), where \(a\) and \(b\) are any numbers, variables, or expressions.

Consider the exercise \((2x+3)^2\). Here, we have the binomial \(2x+3\) being squared, which results in a perfect square trinomial after expansion. Performing the calculations, we start by squaring each term (\