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

S3-SA1-0044

Grade Level:

Class 6

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

Definition

What is it?

(x+a)(x+b) is a special way to multiply two algebraic expressions called binomials. It helps us expand these expressions into a simpler form, usually a trinomial (an expression with three terms). This multiplication follows a simple rule to make calculations easier.

Simple Example

Quick Example

Imagine you have two friends. One friend gives you (a) extra marbles, and the other gives you (b) extra marbles. If you already have (x) marbles, then (x+a) is your total with the first friend, and (x+b) is your total with the second friend. Multiplying them helps you find a combined total in a specific way.

Worked Example

Step-by-Step

Let's expand (x+3)(x+5):

Step 1: Multiply the first terms of each bracket: x * x = x^2
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Step 2: Multiply the outer terms: x * 5 = 5x
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Step 3: Multiply the inner terms: 3 * x = 3x
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Step 4: Multiply the last terms of each bracket: 3 * 5 = 15
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Step 5: Add all these results: x^2 + 5x + 3x + 15
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Step 6: Combine the like terms (5x and 3x): x^2 + (5+3)x + 15
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Step 7: Simplify: x^2 + 8x + 15

Answer: (x+3)(x+5) = x^2 + 8x + 15

Why It Matters

Understanding this multiplication is like learning a secret code for bigger math problems! It's super important for fields like Computer Science to design efficient algorithms and in Engineering to calculate areas or volumes. Future innovators in AI/ML and Data Science use similar logic to build smart systems.

Common Mistakes

MISTAKE: Only multiplying the first terms and the last terms, like (x*x) + (a*b) | CORRECTION: Remember to multiply ALL pairs of terms: first with first, first with second, second with first, and second with second (FOIL method).

MISTAKE: Forgetting to combine the 'like terms' (the terms with 'x') in the middle. | CORRECTION: After multiplying all terms, always look for terms that have the same variable and power (like 3x and 5x) and add or subtract them.

MISTAKE: Making sign errors when 'a' or 'b' are negative. For example, multiplying x * (-5) as 5x instead of -5x. | CORRECTION: Always pay close attention to the positive (+) and negative (-) signs of each term when multiplying.

Practice Questions

Try It Yourself

QUESTION: Expand (y+2)(y+4) | ANSWER: y^2 + 6y + 8

QUESTION: Expand (p+6)(p+1) | ANSWER: p^2 + 7p + 6

QUESTION: If (x+a)(x+b) = x^2 + 9x + 14, what are the values of 'a' and 'b'? (Assume a and b are positive integers) | ANSWER: a=2, b=7 (or a=7, b=2)

MCQ

Quick Quiz

Which of these is the correct expansion of (m+7)(m+2)?

m^2 + 14

m^2 + 9m + 14

m^2 + 7m + 2

m + 9 + 14

The Correct Answer Is:

To expand (m+7)(m+2), we multiply m*m (m^2), m*2 (2m), 7*m (7m), and 7*2 (14). Adding these gives m^2 + 2m + 7m + 14, which simplifies to m^2 + 9m + 14.

Real World Connection

In the Real World

Imagine a farmer in rural India wants to expand his rectangular field. If he increases one side by 'a' meters and the other side by 'b' meters, this algebraic expansion helps him quickly calculate the new total area without drawing or measuring each part separately. This is a basic idea used in architecture and city planning too!

Key Vocabulary

Key Terms

BINOMIAL: An algebraic expression with two terms, like (x+a) | TRINOMIAL: An algebraic expression with three terms, like x^2 + 8x + 15 | EXPANSION: The process of multiplying out brackets to get a longer expression | LIKE TERMS: Terms that have the same variable raised to the same power, like 3x and 5x

What's Next

What to Learn Next

Great job understanding (x+a)(x+b)! Next, you should explore (x-a)(x-b) and (x+a)(x-b). These variations will build on what you've learned and make you even stronger in algebraic identities!