Problem 178 - College Algebra

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 expanded form of the expression \((\sqrt{x+4}+1)^{2}\) is \(x+4+2\sqrt{x+4} +1\).

Step 1 Identify A and B

In the expression \((\sqrt{x+4}+1)^{2}\), \(A\) is represented by \(\sqrt{x+4}\) and \(B\) is represented by \(1\).

Step 2 Apply Special Product Formula

Substitute \(A\) and \(B\) in the special product formula \((A+B)^{2}=A^{2}+2 A B+B^{2}\). So, \((\sqrt{x+4}+1)^{2} = (\sqrt{x+4})^{2}+2(\sqrt{x+4})(1)+(1)^{2}\).

Step 3 Simplify the Result

Simplify each term to get the final expression. \((\sqrt{x+4})^{2} is x+4, 2(\sqrt{x+4})(1)\) simplifies to \(2\sqrt{x+4}\) and \((1)^{2}\) becomes \(1\). Thus, the expanded form will be \(x+4+2\sqrt{x+4} +1\).

Key Concepts

Binomial Expansion

Understanding the concept of binomial expansion is key in algebra. A binomial is an algebraic expression that contains two terms. When you expand a binomial raised to a power, you're essentially multiplying it by itself a number of times indicated by the exponent.
This is often done using the Binomial Theorem, which allows us to express it in a simpler manner. However, for squared binomials, we use the special product formula \[(A + B)^2 = A^2 + 2AB + B^2\].
This formula allows us to expand and simplify the expression without manual multiplication. This process makes solving exercises involving binomials much quicker and reduces computation errors.

• Identify which terms represent \(A\) and \(B\) in the binomial.
• Use the formula to substitute values and simplify the expression.

Practicing these steps will make the expansion process intuitive and quick.

Squaring Binomials

Squaring binomials is a specific instance of binomial expansion where the exponent is 2. When a binomial is squared, like \((A + B)^2\), it implies multiplying the binomial by itself.
Applying the special product formula \[(A + B)^2 = A^2 + 2AB + B^2\] makes this process straightforward. This formula is derived from multiplying the binomial in its entirety and simplifying the terms to achieve a more manageable expression.
In the exercise given, the binomial \((\sqrt{x+4} + 1)^2\) was considered under this concept. Here, \(A\) represents \(\sqrt{x+4}\) and \(B\) is \(1\). By applying the formula, you methodically simplify:

• First term: \(A^2 = (\sqrt{x+4})^2\) results in \(x + 4\).
• Middle term: \(2AB = 2(\sqrt{x+4})(1)\) simplifies to \(2\sqrt{x+4}\).
• Last term: \(B^2 = 1^2\) results in \(1\).

Putting it all together gives a smooth expansion process, fully showcasing the effectiveness of squaring binomials through the special product formula.

Algebraic Expressions

Algebraic expressions form the core of algebra and are made up of constants, variables, and combinations thereof connected by operators like addition and multiplication. They make it possible to generalize mathematical concepts and solve for unknowns.
Expressions like the binomial \((\sqrt{x+4} + 1)^2\) can seem complex, but breaking them down using known formulas such as the special product formula helps in understanding their structure and simplification.
Working with algebraic expressions involves:

• Identifying the parts of the expression — here, \(\sqrt{x+4}\) and \(1\) are parts of the binomial.
• Simplifying and manipulating these parts according to algebraic rules.
• Using formulas and rules to solve or simplify the expressions.

By practicing with various expressions, you become proficient in recognizing patterns and applying formulas efficiently. This skill is crucial as algebra forms the basis for higher-level math and problem-solving in various real-world scenarios.