Algebra as Language Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
Express: 'The square of the sum of aa and bb' vs 'The sum of the squares of aa and bb'. Are they equal?

Solution

  1. 1
    Square of the sum: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2.
  2. 2
    Sum of the squares: a2+b2a^2 + b^2. Not equal! (a+b)2=a2+b2+2ab(a+b)^2 = a^2 + b^2 + 2ab.

Answer

(a+b)2โ‰ a2+b2(a+b)^2 \neq a^2 + b^2 (differ by 2ab2ab).
Precise algebraic language prevents ambiguity. The difference between 'square of the sum' and 'sum of the squares' is exactly 2ab2ab โ€” a common source of errors.

About Algebra as Language

The perspective that algebra is a formal language with syntax and grammar for expressing mathematical ideas and relationships precisely.

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More Algebra as Language Examples

Example 1 easy

Translate into algebra: 'Five more than twice a number is seventeen.'

Example 2 medium

Express in algebra: 'The product of two consecutive integers is 12 more than twice their sum.'

Example 3 easy

Write an equation: 'A number decreased by 7 equals 15.'

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