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Chapter 2: Problem 25

For any values of \(a\) and \(b,\) does \((a+b)^{2}=a^{2}+b^{2} ?\) Why or why not?

Short Answer

Expert verified
No, \((a+b)^2 = a^2 + b^2\) does not hold true for any values of \(a\) and \(b\), except for specific cases where \(2ab = 0\), such as when \(a=0\) or \(b=0\). This is because the expanded and simplified expression is \(a^2 + 2ab + b^2\), which is not equivalent to \(a^2 + b^2\).

Step by step solution

01

Expand the Left Side of the Equation

To expand the left side of the equation, we need to apply the distributive property (also known as the "FOIL" method) to the expression \((a+b)^2\). The "FOIL" method asks you to multiply the First, Outer, Inner, and Last terms of a binomial expression in the following way: \[(a+b)^2 = (a+b)(a+b) = a^2 + ab + ba + b^2\]
02

Simplify the Expanded Expression

Now, simplify the expanded expression, and take note of any common or like terms: \[a^2 + ab + ba + b^2 = a^2 + 2ab + b^2\]
03

Compare the Simplified Expression with the Right Side of the Equation

We now have the simplified expression, \(a^2 + 2ab + b^2\). Compare this expression with the right side of the equation which is \(a^2 + b^2\). We can quickly notice that these expressions are not equal, since the simplified expression has an extra term, \(2ab\), which is not present on the right side of the equation.
04

Determine if the Equation Holds True for Any Values of \(a\) and \(b\)

Since the simplified expression, \(a^2 + 2ab + b^2\), is not equivalent to \(a^2 + b^2\), we can conclude that the equation \((a+b)^2 = a^2 + b^2\) does not hold true for any values of \(a\) and \(b\), except for the specific cases where \(2ab = 0\). In those specific cases, the two sides of the equation will indeed be equal. For example, when \(a=0\) or \(b=0\), the equation holds true: \((0+b)^2=0^2+b^2\) \((a+0)^2=a^2+0^2\) However, for all other values of \(a\) and \(b\), the equation \((a+b)^2 = a^2 + b^2\) does not hold true. Therefore, we can conclude that the given equation is generally false, except for the specific cases mentioned earlier.

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