Problem 19 - Algebra 1

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

A counterexample to the statement \((a + b)^{2} = a^{2} + b^{2}\) is \(a = 1\) and \(b = 1\).

Step 1: Identify a strategy

Since we're looking for a counterexample, the best approach is to try different values of \(a\) and \(b\) and test the equation \((a+b)^{2}=a^{2}+b^{2}\). The simplest numbers to start with would be 1.

Step 2: Apply the formula

Let's plug in \(a = 1\) and \(b = 1\) into both sides of our equation. The left-hand side becomes \((1 + 1)^{2} = 2^{2} = 4\). The right-hand side becomes \(1^{2} + 1^{2} = 1 + 1 = 2\). Here we can see that the left and right sides are not equal, indicating that our selected values are a counterexample to the statement.

Step 3: Confirm the counterexample

Because the left-hand side (\(4\)) is not equal to the right-hand side (\(2\)), we have found a counterexample that disproves the statement. Our counterexample is \(a = 1\) and \(b = 1\).

Key Concepts

Real Numbers

Real numbers encompass all the numbers on the number line. They include various types of numbers such as:

• Natural numbers: Counting numbers like 1, 2, 3, and so on.
• Whole numbers: Natural numbers including zero.
• Integers: Whole numbers that can be positive or negative.
• Rational numbers: Numbers that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q\) is not zero.
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, like \(\pi\) or the square root of 2.

Real numbers can be visualized on a continuous number line, ranging from negative infinity to positive infinity. They are used in almost every aspect of mathematics, from basic arithmetic to complex algebraic calculations. In the context of this exercise, real numbers were used to test a mathematical statement with typical values such as 1.

Quadratic Expressions

Quadratic expressions are algebraic expressions of the form \(ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants and \(x\) is a variable. The term \(ax^2\) makes the expression quadratic since it involves the variable raised to the second power.Quadratic expressions frequently appear in equations like \(y = ax^2 + bx + c\), known for their standard parabolic graph shapes. they describe diverse real-world phenomena such as projectile motion and area calculations.Substituting variables into quadratic expressions helps test mathematical statements, assessing their validity or identifying exceptions.

Equation Verification

Equation verification involves confirming whether both sides of an equation produce the same result for given values. This process is critical in algebra for ensuring that mathematical statements hold true.Verification Process:

• Substitute the given values into each side of the equation.
• Calculate both sides separately to see if they are equal.
• If the results differ, the equation is disproven for those values.

In the exercise provided, equation verification is used to check if \( (a+b)^2 = a^2 + b^2 \) is a valid statement for certain values of \(a\) and \(b\).The failure of the equation when \(a = 1\) and \(b = 1\) highlights the importance of testing different values to confirm a statement's universality.Through this example, we see how crucial equation verification is in identifying counterexamples in algebra.