Problem 178 - College Algebra Essentials

Question

Exercises \(177-179\) will help you prepare for the material covered in the next section. Use the special product \((A+B)^{2}=A^{2}+2 A B+B^{2}\) to multiply: \((\sqrt{x+4}+1)^{2}\)

Step-by-Step Solution

Answer

The result of \((\sqrt{x+4}+1)^2\), when expanded using the perfect square formula, is \(x + 2\sqrt{x+4} + 5\).

Step 1: Identify 'A' and 'B'

According to the given expression, it can be noted that A= \(\sqrt{x+4}\) and B=1.

Step 2: Apply the formula

Now, using the formula \((A + B)^2 = A^2 + 2AB + B^2\), put A=\(\sqrt{x+4}\) and B=1 into the formula to expand the expression. So, \((\sqrt{x+4} + 1)^2 = (\sqrt{x+4})^2 + 2(\sqrt{x+4})(1) + (1)^2\).

Step 3: Simplify the result

Inside the parentheses, \((\sqrt{x+4})^2\), the square root will be removed, which results in \(x+4\). Then \(2(\sqrt{x+4})\) simplifies to \(2\sqrt{x+4}\), and \((1)^2\) simplifies to 1. Therefore, the complete expression simplifies to \(x + 4 + 2\sqrt{x+4} + 1 = x + 2\sqrt{x+4} + 5\).