Inaugurated by IN-SPACe
ISRO Registered Space Tutor
S3-SA1-0297
What is the Identity (a+b+c)²?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The identity (a+b+c)^2 is a special formula in algebra that tells us how to expand the square of the sum of three terms. It means multiplying (a+b+c) by itself. The expanded form is a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.
Simple Example
Quick Example
Imagine you have three friends, A, B, and C. If A has 2 chocolates, B has 3, and C has 4. The total number of chocolates is 2+3+4 = 9. If you want to find the square of this total, it's (2+3+4)^2 = 9^2 = 81. Using the identity helps us find this total in a structured way, by squaring each friend's chocolates and then adding products of pairs.
Worked Example
Step-by-Step
Let's expand (x+2y+3)^2 using the identity (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.
Step 1: Identify a, b, and c. Here, a = x, b = 2y, and c = 3.
---Step 2: Square each term: a^2 = x^2, b^2 = (2y)^2 = 4y^2, c^2 = 3^2 = 9.
---Step 3: Find the product of pairs and multiply by 2: 2ab = 2(x)(2y) = 4xy.
---Step 4: Find the next product of pairs and multiply by 2: 2bc = 2(2y)(3) = 12y.
---Step 5: Find the last product of pairs and multiply by 2: 2ca = 2(3)(x) = 6x.
---Step 6: Add all these parts together: x^2 + 4y^2 + 9 + 4xy + 12y + 6x.
Answer: (x+2y+3)^2 = x^2 + 4y^2 + 9 + 4xy + 12y + 6x.
Why It Matters
This identity is super useful for simplifying complex math problems, especially in fields like Engineering and Computer Science where calculations involving many variables are common. It helps engineers design structures efficiently and computer scientists optimize algorithms. Even in Data Science, understanding how terms interact when squared helps in analyzing patterns.
Common Mistakes
MISTAKE: Forgetting the '2' in front of the product terms (e.g., writing ab instead of 2ab). | CORRECTION: Remember that when expanding (a+b+c)^2, each pair of terms is multiplied by 2.
MISTAKE: Incorrectly squaring negative terms (e.g., (-2)^2 = -4). | CORRECTION: Always remember that the square of any negative number is positive (e.g., (-2)^2 = 4).
MISTAKE: Not squaring the coefficient when a term has one (e.g., (2y)^2 = 2y^2). | CORRECTION: Square both the coefficient and the variable (e.g., (2y)^2 = 2^2 * y^2 = 4y^2).
Practice Questions
Try It Yourself
QUESTION: Expand (p+q+r)^2. | ANSWER: p^2 + q^2 + r^2 + 2pq + 2qr + 2rp
QUESTION: Expand (2x+y+1)^2. | ANSWER: 4x^2 + y^2 + 1 + 4xy + 2y + 4x
QUESTION: If a+b+c = 10 and a^2+b^2+c^2 = 30, find the value of ab+bc+ca. | ANSWER: We know (a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca). So, (10)^2 = 30 + 2(ab+bc+ca). 100 = 30 + 2(ab+bc+ca). 70 = 2(ab+bc+ca). Therefore, ab+bc+ca = 35.
MCQ
Quick Quiz
Which of the following is the correct expansion of (x-y+z)^2?
x^2 + y^2 + z^2 - 2xy + 2yz + 2zx
x^2 + y^2 + z^2 - 2xy - 2yz + 2zx
x^2 - y^2 + z^2 - 2xy - 2yz + 2zx
x^2 + y^2 + z^2 - 2xy - 2yz - 2zx
The Correct Answer Is:
B
When we apply the identity (a+b+c)^2, we treat -y as 'b'. So 2ab becomes 2(x)(-y) = -2xy. 2bc becomes 2(-y)(z) = -2yz. 2ca becomes 2(z)(x) = 2zx. And (-y)^2 is y^2.
Real World Connection
In the Real World
Imagine an architect designing a building with three different sections. To calculate the total surface area for painting or tiling, if the dimensions of each section are represented by a, b, and c, this identity helps simplify the calculations quickly. For example, in construction planning in cities like Bengaluru, complex area calculations are common, and these identities make them much faster.
Key Vocabulary
Key Terms
IDENTITY: A mathematical equation that is true for all possible values of the variables involved. | EXPANSION: The process of multiplying out the terms in an expression. | COEFFICIENT: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g., 2 in 2x). | VARIABLE: A symbol, usually a letter, representing a quantity that may vary.
What's Next
What to Learn Next
Great job understanding (a+b+c)^2! Next, you can explore other algebraic identities like (a+b)^3 or (a-b)^3. These identities build on what you've learned and will help you solve even more complex problems in higher classes!
