Calculate Rectangle Area: Finding Area When Length is (8a-b) and Width is (2a+3b)

Q: Why can't I just add the side lengths together?

That would give you the perimeter, not the area! Area requires multiplication of length × width, while perimeter uses addition of all sides.

Q: How do I remember which terms to multiply together?

Use the FOIL method: First terms, Outer terms, Inner terms, Last terms. For (8a-b)(2a+3b): 8a×2a, 8a×3b, -b×2a, -b×3b.

Q: What if I get confused with the negative signs?

Write it step by step! When you see -b × 2a, remember that negative times positive equals negative: -b × 2a = -2ab.

Q: How do I combine like terms at the end?

Look for terms with the same variables and exponents. Here, 24ab and -2ab are like terms: 24ab - 2ab = 22ab.

Q: Can I check my answer somehow?

Yes! Try substituting simple values like a=1, b=1 into both the original expression (8a-b)(2a+3b) and your answer. They should give the same result!

Polynomial Multiplication with Binomial Expressions

Calculate the area of the rectangle in the diagram and express it in terms of a and b.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the area of the rectangle using A,B

00:04 We'll use the formula for calculating rectangle area (side times side)

00:08 We'll substitute appropriate values according to the given data and solve for the area

00:16 We'll properly expand the brackets, multiply each term by each term

00:34 We'll calculate the products

00:51 We'll collect like terms

01:00 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Calculate the area of the rectangle in the diagram and express it in terms of a and b.

Step-by-step solution

To solve this problem, we need to calculate the area of the rectangle with side lengths $(8a-b)$ and $(2a+3b)$ .

The area $A$ is found by multiplying these two expressions:

Step 1: Write the expression for the area:
$A = (8a-b)(2a+3b)$
Step 2: Use the distributive property to expand the product:
$A = 8a(2a) + 8a(3b) - b(2a) - b(3b)$ .
Step 3: Calculate each term individually:
- $8a \times 2a = 16a^2$
- $8a \times 3b = 24ab$
- $-b \times 2a = -2ab$
- $-b \times 3b = -3b^2$
Step 4: Combine like terms:
$A = 16a^2 + 24ab - 2ab - 3b^2$ , which simplifies to $16a^2 + 22ab - 3b^2$ .

Therefore, the area of the rectangle, expressed in terms of $a$ and $b$ , is $16a^2 + 22ab - 3b^2$ .

Final Answer

$16a^2+22ab-3b^2$

Key Points to Remember

Essential concepts to master this topic

Area Formula: Rectangle area equals length times width
FOIL Method: (8a-b)(2a+3b) = 16a² + 24ab - 2ab - 3b²
Check: Combine like terms: 24ab - 2ab = 22ab ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute negative signs
Don't just multiply 8a by (2a+3b) and forget about -b = missing terms! The negative sign applies to both terms when distributing. Always distribute each term in the first binomial to each term in the second binomial.

Practice Quiz

Test your knowledge with interactive questions

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

$$ (ab)(c d) $$

$$

FAQ

Everything you need to know about this question

Why can't I just add the side lengths together?

That would give you the perimeter, not the area! Area requires multiplication of length × width, while perimeter uses addition of all sides.

How do I remember which terms to multiply together?

Use the FOIL method: First terms, Outer terms, Inner terms, Last terms. For (8a-b)(2a+3b): 8a×2a, 8a×3b, -b×2a, -b×3b.

What if I get confused with the negative signs?

Write it step by step! When you see -b × 2a, remember that negative times positive equals negative: -b × 2a = -2ab.

How do I combine like terms at the end?

Look for terms with the same variables and exponents. Here, 24ab and -2ab are like terms: 24ab - 2ab = 22ab.

Can I check my answer somehow?

Yes! Try substituting simple values like a=1, b=1 into both the original expression (8a-b)(2a+3b) and your answer. They should give the same result!

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