What is (x+a)(x+b) Product?

S3-SA1-0752

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The product (x+a)(x+b) is a special way to multiply two algebraic expressions. It helps us find the result when two terms, each having 'x' and a constant number, are multiplied together. This multiplication expands into a specific pattern, which is x^2 + (a+b)x + ab.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, buying samosas. Rohan buys (x+2) samosas and Priya buys (x+3) samosas. If you want to know the total 'product' of their samosa buying patterns, you would multiply (x+2) by (x+3).

Worked Example

Step-by-Step

Let's find the product of (x+5)(x+3).

Step 1: Identify 'a' and 'b'. Here, a = 5 and b = 3.

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Step 2: Recall the formula: (x+a)(x+b) = x^2 + (a+b)x + ab.

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Step 3: Substitute the values of 'a' and 'b' into the formula.
x^2 + (5+3)x + (5*3)

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Step 4: Simplify the terms.
x^2 + (8)x + (15)

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Step 5: Write the final product.
x^2 + 8x + 15

Answer: The product of (x+5)(x+3) is x^2 + 8x + 15.

Why It Matters

Understanding this product helps in solving complex problems in various fields. Engineers use it to design bridges and buildings, while computer scientists apply similar logic in programming and developing algorithms. Even in data science, understanding such expansions can help in analyzing trends and making predictions.

Common Mistakes

MISTAKE: Forgetting to multiply 'ab' at the end, only doing x^2 + (a+b)x | CORRECTION: Remember that the last term is the product of 'a' and 'b', so include 'ab' in your final answer.

MISTAKE: Incorrectly adding 'a' and 'b' (e.g., if one is negative) | CORRECTION: Pay close attention to the signs of 'a' and 'b'. If a = 5 and b = -2, then (a+b) = 5 + (-2) = 3.

MISTAKE: Not multiplying 'x' by (a+b), just writing x^2 + a+b + ab | CORRECTION: The middle term is (a+b) multiplied by 'x', so it should be (a+b)x.

Practice Questions

Try It Yourself

QUESTION: Find the product of (x+4)(x+6). | ANSWER: x^2 + 10x + 24

QUESTION: What is the product of (x+7)(x-2)? | ANSWER: x^2 + 5x - 14

QUESTION: If a rectangular field has a length of (x+9) meters and a width of (x+1) meters, what is its area? | ANSWER: x^2 + 10x + 9 square meters

MCQ

Quick Quiz

Which of the following is the correct expansion of (x+2)(x+5)?

x^2 + 7x + 7

x^2 + 10x + 7

x^2 + 7x + 10

x^2 + 10x + 10

The Correct Answer Is:

Using the formula x^2 + (a+b)x + ab, with a=2 and b=5, we get x^2 + (2+5)x + (2*5) which simplifies to x^2 + 7x + 10.

Real World Connection

In the Real World

Imagine a business like a small kirana store in your neighbourhood. If the store owner wants to calculate potential profits where daily sales 'x' increase by a certain amount 'a' and 'b' for two different products, they might use similar algebraic thinking. This helps them plan inventory or estimate earnings, just like how e-commerce platforms use such calculations to optimize delivery routes or pricing.

Key Vocabulary

Key Terms

PRODUCT: The result obtained when two or more numbers or expressions are multiplied together. | EXPRESSION: A combination of numbers, variables, and operation symbols. | VARIABLE: A symbol, usually a letter like 'x', representing a quantity that can change. | CONSTANT: A value that does not change. | EXPANSION: The process of writing an algebraic expression as a sum of terms.

What's Next

What to Learn Next

Great job understanding this basic product! Next, you can explore other algebraic identities like (a+b)^2 or (a-b)^2. These build directly on what you've learned here and will make solving more complex equations even easier and faster!