(10+x)^2 and 100 + x^2

Question

Show the following expressions are not equivalent in 2 different ways; (10 +x)^2 and 100+x^2

Answer 1

1. Algebraically:  (10+x)² is equal to 100 + 20x + x2.  That is not the same as 100 + x², since they differ by 20x.  As long as x is not 0, the expressions will be different.

 

2. By counterexample:  Try it when x = 1.  (10+x)² = (10 + 1)² = 11² = 121.  But 100 + x² = 100 + 1 = 101.  As long as the expressions are not equal for at least one case, then they can't be equivalent.