Problem 47 - Algebra 1: Concepts and Skills

Question

Tell whether the statement is true or false. If the statement is false, rewrite the right-hand side to make the statement true. $$ (a+2 b)^{2}=a^{2}+2 a b+4 b^{2} $$

Step-by-Step Solution

Answer

The statement is false. The correct version is \( (a+2b)^2 = a^2 + 4ab + 4b^2 \).

Step 1: Expansion of Left Hand Side

Expand the left-hand side by squaring the binomial (a+2b)^2. This becomes \( (a+2b) * (a+2b) \) which simplifies to \( a^2 + 2ab + 2ab + 4b^2 \) or \( a^2 + 4ab + 4b^2 \)

Step 2: Comparison with Right Hand Side

Once the left-hand side is fully expanded, we realize that it equals \( a^2 + 4ab + 4b^2 \), which does not match the right-hand side \( a^2 + 2ab + 4b^2 \) given in the statement.

Step 3: Correction of Right Hand Side

As the right hand side of the statement does not match our calculated left hand side, it needs correction. Seeing that the term 2ab on the right should be 4ab, the correct version of the statement therefore becomes \( (a+2b)^2 = a^2 + 4ab + 4b^2 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols, such as addition, subtraction, multiplication, and division. They are the building blocks of algebra and allow us to express quantitative relationships in a symbolic form. For instance, in the expression \(a + 2b\), \(a\) and \(b\) are variables representing quantities that can vary, while 2 is a constant multiplier of \(b\).

An expression becomes an equation when it includes an equals sign. Equations can often be solved to find the value of the variables. The beauty of algebraic expressions lies in their ability to represent complex relationships in a compact and understandable way, making them a fundamental concept in mathematics and essential for solving real-world problems.

Squaring Binomials

Squaring binomials is a specific case of multiplying binomials, where both binomials are identical. A binomial is simply an algebraic expression with two terms, for example, \(a + b\) or \(a - b\). When you square a binomial, say \(a + b\), you are multiplying it by itself: \(a + b)^2 = (a + b) (a + b)\).

Expanding this using the distributive property, you get \(a^2 + ab + ba + b^2\). Since \(ab\) and \(ba\) are the same, they can be combined to \(2ab\), resulting in \(a^2 + 2ab + b^2\). This is a fundamental pattern in algebra called the square of a sum. Similarly, the square of a difference \(a - b)^2\) would expand to \(a^2 - 2ab + b^2\). Remembering these patterns helps in simplifying and calculating expressions efficiently.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. It involves a sequence of logical deductions from known facts or accepted truths, called axioms, and previously proved statements or theorems. In the context of algebraic expressions, a proof might involve showing that two expressions are equivalent by expanding, simplifying, and manipulating them according to mathematical rules.

In our exercise, the proof process involved expanding the left-hand side and comparing it with the right-hand side to show whether the original statement \( (a + 2b)^2 = a^2 + 2ab + 4b^2 \) was true. Upon finding the left-hand side to be \( a^2 + 4ab + 4b^2 \), we proved that the original equation was incorrect and then corrected it. This process of expansion, simplification, and comparison is an example of a mathematical proof in action, showcasing how we can validate or correct algebraic equations.