What is (x+a)(x+b) Identity? | Simple Explanation for Class 10

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What is the Identity (x+a)(x+b)?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The identity (x+a)(x+b) is a special algebraic formula that helps us multiply two binomials (expressions with two terms) quickly. It states that (x+a)(x+b) is always equal to x^2 + (a+b)x + ab. This formula is true for any values of x, a, and b.

Simple Example

Quick Example

Imagine you want to find the area of a rectangular field. If one side is (x+3) meters and the other side is (x+5) meters, you can use this identity to find the total area. Here, a=3 and b=5. So the area would be x^2 + (3+5)x + (3*5) = x^2 + 8x + 15 square meters.

Worked Example

Step-by-Step

Let's expand (y+4)(y+7) using the identity (x+a)(x+b) = x^2 + (a+b)x + ab.
---Step 1: Identify x, a, and b. In our problem, x = y, a = 4, and b = 7.
---Step 2: Substitute these values into the formula x^2 + (a+b)x + ab.
---Step 3: Replace x with y: y^2 + (a+b)y + ab.
---Step 4: Substitute a=4 and b=7: y^2 + (4+7)y + (4*7).
---Step 5: Perform the additions and multiplications: y^2 + (11)y + (28).
---Step 6: Simplify the expression: y^2 + 11y + 28.
Answer: (y+4)(y+7) = y^2 + 11y + 28.

Why It Matters

This identity helps engineers quickly calculate areas or volumes in design, and is crucial in computer programming for optimizing calculations. In fields like AI/ML, similar algebraic expansions are used to simplify complex equations, making computations faster and more efficient. It's a foundational skill for anyone working with data or complex systems.

Common Mistakes

MISTAKE: Forgetting the middle term or incorrectly adding/multiplying a and b. Forgetting (a+b)x or mixing up addition and multiplication. | CORRECTION: Remember the full formula: x^2 + (a+b)x + ab. Always add 'a' and 'b' for the middle term's coefficient, and multiply 'a' and 'b' for the last term.

MISTAKE: Applying the identity when the terms are not in the (x+a)(x+b) format, for example, (2x+a)(x+b). | CORRECTION: Ensure the 'x' term in both binomials is exactly the same and has a coefficient of 1. If not, you might need to factor out a common term first or use general binomial multiplication.

MISTAKE: Making sign errors when 'a' or 'b' are negative. Forgetting that a negative times a negative is a positive. | CORRECTION: Treat 'a' and 'b' with their signs. For example, in (x-3)(x+5), a=-3 and b=5. So (a+b) = -3+5 = 2, and (ab) = -3*5 = -15.

Practice Questions

Try It Yourself

QUESTION: Expand (p+6)(p+2) using the identity. | ANSWER: p^2 + 8p + 12

QUESTION: Expand (k-5)(k+9) using the identity. Be careful with signs! | ANSWER: k^2 + 4k - 45

QUESTION: If (m+7)(m-3) = m^2 + Am + B, find the values of A and B. | ANSWER: A=4, B=-21

MCQ

Quick Quiz

Which of the following is the correct expansion of (x+8)(x+1)?

x^2 + 9x + 8

x^2 + 8x + 9

x^2 + 7x + 8

x^2 + 9x + 9

The Correct Answer Is:

Using the identity x^2 + (a+b)x + ab, with a=8 and b=1, we get x^2 + (8+1)x + (8*1), which simplifies to x^2 + 9x + 8.

Real World Connection

In the Real World

Imagine an architect designing a building. They might use expressions like (length + extension) * (width + extension) to calculate new areas. This identity helps them quickly find the total area without doing long multiplication, saving time and reducing errors in their designs for new apartments or shopping malls.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables | BINOMIAL: An algebraic expression with exactly two terms, like (x+a) | EXPANSION: The process of multiplying out brackets to write an expression as a sum of terms | COEFFICIENT: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g., 5 in 5x) | VARIABLE: A symbol, usually a letter, representing a quantity that may vary (e.g., x or y)

What's Next

What to Learn Next

Great job understanding this identity! Next, you should explore other important algebraic identities like (a+b)^2 and (a-b)^2. These build directly on what you've learned and are super useful for solving even more complex problems in algebra and beyond.