Problem 13 - Algebra 1: Concepts and Skills

Question

Find a counterexample to show that the statement is not true. If \(a\) and \(b\) are real numbers, then \((a+b)^{2}=a^{2}+b^{2}\)

Step-by-Step Solution

Answer

The chosen real numbers 'a' = 2 and 'b' = 3 act as counterexamples to the given statement as \((a+b)^{2}\) (25) is not equal to \(a^{2} + b^{2}\) (13).

Step 1: Select Real Numbers 'a' and 'b'

Let's choose real numbers 'a' and 'b'. For instance, take \(a = 2\) and \(b = 3\).

Step 2: Apply the Binomial Theorem

Use the property from the binomial theorem that states \((a+b)^{2} = a^{2} + 2ab + b^{2}\) to find \((a+b)^{2} = (2 + 3)^{2} = 25\).

Step 3: Calculate \( a^{2} + b^{2} \)

Calculate \( a^{2} + b^{2} = 2^{2} + 3^{2} = 4 + 9 = 13 \).

Step 4: Compare the Results

Compare the results from Step 2 and Step 3. Since 25 ≠ 13, \((a+b)^{2} ≠ a^{2} + b^{2}\). Hence, 'a' and 'b' are the counterexamples to the given statement.