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Welcome to our Math lesson on Expanding Expressions of the Form (a + b + c)2, this is the ninth lesson of our suite of math lessons covering the topic of Special Algebraic Identities Obtained through Expanding, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Expanding Expressions of the Form (a + b + c)2
Expanding this type of expression involves two steps, in the first step we consider the first two variables as a whole. Then, we continue with the rest. In this algebraic expression we have to use the first identity (a + b)2 = a2 + 2ab + b2 twice. We have
(a + b + c)2 = [(a + b) + c]2
= (a + b)2 + 2 ∙ (a + b) ∙ c + c2
= a2 + 2ab + b2 + 2ac + 2bc + c2
= a2 + b2 + c2 + 2ab + 2ac + 2bc
In this way, we obtain the eighth special algebraic identity
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
Example 8
A triangle has its sides in a ratio 2 : 3 : 4. What is the area of a square whose side length is equal to the perimeter of the triangle? Write the answer in terms of the constant k of the ratio.
Solution 8
We can express the sides of the triangle as a, b and c respectively. Hence, its perimeter is P = a + b + c. This also corresponds to the square side. Since the area A of square is obtained by raising its side length in the power two, we have
A = P2 = (a + b + c)2
= a2 + b2 + c2 + 2ab + 2ac + 2bc
Since a = 2k, b = 3k and c = 4k, we obtain
A = (2k)2 + (3k)2 + (4k)2 + 2 ∙ 2k ∙ 3k + 2 ∙ 2k ∙ 4k + 2 ∙ 3k ∙ 4k
= 4k2 + 9k2 + 16k2 + 12k2 + 16k2 + 24k2
= 81k2
This result could have been obtained by calculating the perimeter of triangle in term of k first (P = 2k + 3 k + 4 k = 9k) and then raising it by the power of two to calculate the area of square, i.e. (9k)2 = 81k2. As you see, the result is the same in both cases, so the method we used to solve this problem is confirmed as correct.
More Special Algebraic Identities Obtained through Expanding Lessons and Learning Resources
Expressions Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 6.3 | Special Algebraic Identities Obtained through Expanding | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 6.3.1 | The Meaning of a Binomial | | |
| 6.3.2 | Square of a Sum | | |
| 6.3.3 | Square of a Difference | | |
| 6.3.4 | Product of Conjugates | | |
| 6.3.5 | Cube of a Sum | | |
| 6.3.6 | Cube of a Difference | | |
| 6.3.7 | Expanding the Algebraic Expression of the Form (a - b) · (a2 + ab + b2) | | |
| 6.3.8 | Expanding the Algebraic Expression of the Form (a + b) · (a2 - ab + b2) | | |
| 6.3.9 | Expanding Expressions of the Form (a + b + c)2 | | |
| 6.3.10 | Combining Special Identities | | |
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