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What is the Identity (a+b+c)^2?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity (a+b+c)^2 is a formula that helps us expand the square of the sum of three terms. It states that when you multiply (a+b+c) by itself, the result is a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.
Simple Example
Quick Example
Imagine you have three friends, A, B, and C, and each contributes some money for a picnic. If you want to calculate the square of their total contribution, (A+B+C)^2, this identity helps you do it quickly. It breaks down the total into individual squares and all possible pairs of products.
Worked Example
Step-by-Step
Let's expand (x + 2y + 3)^2
---Step 1: Identify a, b, and c. Here, a = x, b = 2y, and c = 3.
---Step 2: Apply the identity (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.
---Step 3: Substitute the values into the formula.
(x)^2 + (2y)^2 + (3)^2 + 2(x)(2y) + 2(2y)(3) + 2(3)(x)
---Step 4: Calculate the squares.
x^2 + 4y^2 + 9
---Step 5: Calculate the product terms.
+ 4xy + 12y + 6x
---Step 6: Combine all terms.
x^2 + 4y^2 + 9 + 4xy + 12y + 6x
Answer: x^2 + 4y^2 + 9 + 4xy + 12y + 6x
Why It Matters
Understanding this identity is crucial in fields like Engineering and Physics for simplifying complex equations, especially when dealing with areas or volumes in 3D space. It's also used in AI/ML algorithms for optimizing calculations and in Chemistry for solving certain molecular structure problems, helping scientists and engineers design better products and understand complex systems.
Common Mistakes
MISTAKE: Forgetting the '2' in the product terms (e.g., writing ab instead of 2ab) | CORRECTION: Remember that when you multiply (a+b+c) by (a+b+c), each pair of terms (like 'a' and 'b') appears twice (once as 'ab' and once as 'ba'), so they sum up to 2ab.
MISTAKE: Squaring only the variable part of a term like (2y)^2 and writing 2y^2 instead of 4y^2 | CORRECTION: When a term like '2y' is squared, both the number and the variable are squared: (2y)^2 = 2^2 * y^2 = 4y^2.
MISTAKE: Not including all three possible pair products (ab, bc, ca) | CORRECTION: Always ensure you have six terms in total after expansion: three squared terms (a^2, b^2, c^2) and three doubled product terms (2ab, 2bc, 2ca).
Practice Questions
Try It Yourself
QUESTION: Expand (p + q + r)^2 | ANSWER: p^2 + q^2 + r^2 + 2pq + 2qr + 2rp
QUESTION: Expand (2a + b + 5)^2 | ANSWER: 4a^2 + b^2 + 25 + 4ab + 10b + 20a
QUESTION: If x + y + z = 10 and xy + yz + zx = 20, find the value of x^2 + y^2 + z^2. | ANSWER: We know (x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx). Substituting the given values: (10)^2 = x^2 + y^2 + z^2 + 2(20). So, 100 = x^2 + y^2 + z^2 + 40. Therefore, x^2 + y^2 + z^2 = 100 - 40 = 60.
MCQ
Quick Quiz
Which of the following is the correct expansion of (m - n + p)^2?
m^2 + n^2 + p^2 - 2mn + 2np + 2mp
m^2 + n^2 + p^2 - 2mn - 2np + 2mp
m^2 + n^2 + p^2 + 2mn + 2np + 2mp
m^2 - n^2 + p^2 - 2mn + 2np + 2mp
The Correct Answer Is:
B
Let a=m, b=-n, c=p. Using the identity a^2+b^2+c^2+2ab+2bc+2ca, we get m^2+(-n)^2+p^2+2(m)(-n)+2(-n)(p)+2(p)(m), which simplifies to m^2+n^2+p^2-2mn-2np+2mp.
Real World Connection
In the Real World
This identity is used in computer graphics to calculate distances and transformations in 3D space, which helps in designing video games or creating special effects in movies. Engineers use it when calculating the surface area of complex shapes in construction or the spread of signals in telecommunications, like designing mobile towers for better network coverage across different regions in India.
Key Vocabulary
Key Terms
IDENTITY: An equation that is true for all values of its variables | EXPAND: To multiply out the terms in an expression | TERM: A single number, variable, or product/quotient of numbers and variables | SQUARE: The result of multiplying a number or expression by itself | COEFFICIENT: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.
What's Next
What to Learn Next
Next, you can explore the identities for (a+b)^3 and (a-b)^3. These build on your understanding of expanding expressions and are super useful for solving more complex polynomial problems and even in higher mathematics.
