Question

What is another way of writing \((a+b)(a-b) ?\) (A) \(a^{2}-b^{2} \quad\) OR (B) \(a^{2}+b^{2}\)

Short answer

The expression is equivalent to (A) \(a^2-b^2\).

Step-by-step solution

Understand the problem

We are given the expression \((a+b)(a-b)\) and need to identify its equivalent form from two given options: \(a^{2}-b^{2}\) and \(a^{2}+b^{2}\).

Apply the difference of squares formula

Recall the algebraic identity for the difference of squares: \((x+y)(x-y) = x^2 - y^2\). In our problem, \(x\) is \(a\) and \(y\) is \(b\), so we use this identity.

Substitute variables into the formula

Substitute \(a\) for \(x\) and \(b\) for \(y\) in the difference of squares formula, giving \((a+b)(a-b) = a^2 - b^2\).

Compare with given options

The equivalent expression for \((a+b)(a-b)\) is \(a^2 - b^2\), which matches option (A). Option (B), \(a^2 + b^2\), does not match our derived expression.

Key concepts

Algebraic Identities

Algebraic identities are powerful tools in mathematics that allow us to simplify expressions and solve equations. These identities are true for all values of the variables involved. One of the most important algebraic identities is the **difference of squares**. The formula for the difference of squares is given by

• \((x + y)(x - y) = x^2 - y^2\)

This identity shows that multiplying the sum and difference of the same two numbers simplifies to the square of the first number minus the square of the second number. This identity is useful when solving quadratic equations or simplifying complex algebraic expressions, as it decreases the number of terms and makes calculations easier. When applying this identity, recognizing the pattern is key: look for expressions that are products of binomials with terms that are identical except for the sign between them, like \((a + b)(a - b)\).
Recognizing and using algebraic identities like the difference of squares can drastically simplify algebraic problem solving.

Equivalent Expressions

Equivalent expressions are expressions that may look different but have the same value for all values of the involved variables. In algebra, finding equivalent expressions is often about transforming one expression into another form using rules and identities.

• For example, the expression \((a + b)(a - b)\) is equivalent to \(a^2 - b^2\) by the difference of squares rule.
• It's crucial to recognize that although these expressions appear differently, they produce identical results when evaluated.

By manipulating and reducing expressions to their simplest forms, students can better understand the underlying structure and relationships between algebraic terms. This process broadens mathematical thinking and enhances flexibility in problem-solving. In practice, always check equivalency by substituting specific values for variables to see if both expressions yield the same result, reaffirming their equivalence.

Problem Solving in Mathematics

Problem solving in mathematics involves a set of strategies tailored to simplify and resolve mathematical questions. To tackle problems efficiently, it is often beneficial to understand broader concepts before diving into specific algebraic manipulations.

• Start by ensuring you comprehend the problem requirements, as seen in the provided exercise where identifying an equivalent expression was key.
• Next, apply appropriate mathematical identities or rules. For \((a+b)(a-b)\) , using the difference of squares simplifies finding the solution.
• Once you apply the formula, ensure that your conclusions match the question's requirements by verifying your final results.

This strategic approach minimizes mistakes and improves efficiency. Additionally, employing steps like these encourages logical reasoning, allows for self-checking of answers, and builds a foundation for tackling more advanced mathematics. When solving any problem, review similar problems or algebraic identities to reinforce your understanding and ability to draw connections across different areas of math.