Expand (a+3b)²: Converting to Multiplication and Addition Forms

Q: Why is the middle term 6ab and not 3ab?

The middle term comes from 2xy in the formula. Since x = a and y = 3b, we get $ 2(a)(3b) = 6ab $. Don't forget to multiply by 2!

Q: How do I remember the binomial square formula?

Think "First squared, plus twice the product, plus second squared". For $ (a+3b)^2 $: first² + 2(first)(second) + second²

Q: Can I just multiply (a+3b)(a+3b) using FOIL instead?

Absolutely! FOIL gives the same result: First: $ a \cdot a = a^2 $Outer: $ a \cdot 3b = 3ab $Inner: $ 3b \cdot a = 3ab $Last: $ 3b \cdot 3b = 9b^2 $Combined: $ a^2 + 6ab + 9b^2 $

Q: What's the difference between the multiplication and addition forms?

The multiplication form shows the original expression as a product: $ (a+3b)(a+3b) $. The addition form shows it expanded: $ a^2 + 6ab + 9b^2 $. Same value, different representations!

Q: How do I know when to square the coefficient?

Always square everything in each term! For $ (3b)^2 $, square both the coefficient 3 and the variable b: $ 3^2 \cdot b^2 = 9b^2 $.

Binomial Expansion with Perfect Square Trinomials

Rewrite the following expression as a multiplication and as an addition:

$(a+3b)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express as a sum and product

00:10 A squared term is actually a multiplication of itself by itself

00:25 We'll use the short multiplication formulas to expand the brackets

00:35 In our exercise A is the X

00:42 and 3B is the Y

00:47 We'll substitute according to the formula and solve

00:57 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

Rewrite the following expression as a multiplication and as an addition:

$(a+3b)^2$

Step-by-step solution

To solve this problem, we need to express $(a+3b)^2$ in two forms: as a multiplication of like terms and as an addition of polynomial terms.

Step 1: Express as a multiplication
The expression $(a+3b)^2$ can be rewritten as $(a+3b)(a+3b)$ . This indicates the binomial is multiplied by itself.
Step 2: Expand using the binomial theorem
We apply the formula for squaring a binomial:
$(x+y)^2 = x^2 + 2xy + y^2$ .
Here, let $x = a$ and $y = 3b$ . Applying the formula we get:
$(a+3b)^2 = a^2 + 2(a)(3b) + (3b)^2$ .
Step 3: Simplify
Perform the calculations for each term:
- $a^2$ : The square of $a$ .
- $2(a)(3b) = 6ab$ : Multiply the terms and the coefficients.
- $(3b)^2 = 9b^2$ : Square the coefficient and the variable.
Combining these, the expanded form is:
$a^2 + 6ab + 9b^2$ .

Thus, the expression as a multiplication is $(a+3b)(a+3b)$ , and as an addition, it is $a^2 + 6ab + 9b^2$ .

Therefore, the solution to the problem is $(a+3b)(a+3b)$ and $a^2+6ab+9b^2$ .

Final Answer

$(a+3b)(a+3b)$

$a^2+6ab+9b^2$

Key Points to Remember

Essential concepts to master this topic

Formula: $(x+y)^2 = x^2 + 2xy + y^2$ for all binomial squares
Technique: Identify x = a and y = 3b, then calculate $2(a)(3b) = 6ab$
Check: Verify by multiplying $(a+3b)(a+3b)$ using FOIL method ✓

Common Mistakes

Avoid these frequent errors

Forgetting the middle term when expanding
Don't write $(a+3b)^2 = a^2 + 9b^2$ without the middle term! This ignores the 2xy part of the formula and gives an incomplete answer. Always include all three terms: $a^2 + 6ab + 9b^2$ .

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+y)^2 $$

FAQ

Everything you need to know about this question

Why is the middle term 6ab and not 3ab?

The middle term comes from 2xy in the formula. Since x = a and y = 3b, we get $2(a)(3b) = 6ab$ . Don't forget to multiply by 2!

How do I remember the binomial square formula?

Think "First squared, plus twice the product, plus second squared". For $(a+3b)^2$ : first² + 2(first)(second) + second²

Can I just multiply (a+3b)(a+3b) using FOIL instead?

Absolutely! FOIL gives the same result:

First: $a \cdot a = a^2$
Outer: $a \cdot 3b = 3ab$
Inner: $3b \cdot a = 3ab$
Last: $3b \cdot 3b = 9b^2$

Combined:

a^2 + 6ab + 9b^2

What's the difference between the multiplication and addition forms?

The multiplication form shows the original expression as a product: $(a+3b)(a+3b)$ . The addition form shows it expanded: $a^2 + 6ab + 9b^2$ . Same value, different representations!

How do I know when to square the coefficient?

Always square everything in each term! For $(3b)^2$ , square both the coefficient 3 and the variable b: $3^2 \cdot b^2 = 9b^2$ .

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