Expand (3x+4)²: Perfect Square Binomial Expression

Q: Why can't I just square each term separately?

Squaring $ (3x + 4) $ means multiplying the entire expression by itself, not just squaring individual terms. You get: $ (3x + 4)(3x + 4) $, which includes cross-terms when you distribute!

Q: How do I remember the perfect square formula?

Think "First squared, twice the product, last squared": $ (a + b)^2 = a^2 + 2ab + b^2 $. The middle term $ 2ab $ comes from multiplying each term by the other!

Q: What if I have a negative sign like (3x - 4)²?

Use the same formula but be careful with signs! $ (a - b)^2 = a^2 - 2ab + b^2 $. The middle term becomes negative because you're subtracting.

Q: Can I use FOIL instead of the formula?

Absolutely! FOIL gives the same result: $ (3x + 4)(3x + 4) $ = First + Outer + Inner + Last. The perfect square formula is just a shortcut for this special case.

Q: How do I check if my expansion is correct?

Pick a simple number like $ x = 1 $. Calculate $ (3(1) + 4)^2 = 7^2 = 49 $, then check your expansion: $ 9(1)^2 + 24(1) + 16 = 9 + 24 + 16 = 49 $ ✓

Perfect Square Expansion with Binomial Formula

$(3x+4)^2=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve using shortened multiplication formulas

00:03 We will use shortened multiplication formulas to expand the brackets

00:12 In our exercise 3X is A

00:19 and 4 is B

00:23 Let's substitute according to the formula

00:31 Let's calculate the squares and products

00:50 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

$(3x+4)^2=\text{?}$

Step-by-step solution

To solve the problem $(3x + 4)^2$ , we will use the formula for the square of a binomial:

$(a + b)^2 = a^2 + 2ab + b^2$

Identify $a$ and $b$ : Here, $a = 3x$ and $b = 4$ .
Apply the formula:

1. Calculate $a^2$ which is $(3x)^2 = 9x^2$ .

2. Calculate $2ab$ which is $2 \times (3x) \times 4 = 24x$ .

3. Calculate $b^2$ which is $4^2 = 16$ .

Combine the results:

$a^2 + 2ab + b^2 = 9x^2 + 24x + 16$ .

Therefore, the expanded form of $(3x + 4)^2$ is $9x^2 + 24x + 16$ .

The correct answer choice is: $9x^2 + 24x + 16$ .

Final Answer

$9x^2+24x+16$

Key Points to Remember

Essential concepts to master this topic

Formula: $(a + b)^2 = a^2 + 2ab + b^2$
Technique: Calculate each term: $(3x)^2 = 9x^2$ , $2(3x)(4) = 24x$ , $4^2 = 16$
Check: Verify by expanding manually or substituting test value: $x=1$ gives $(7)^2 = 49$ and $9+24+16 = 49$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the middle term 2ab
Don't just square each term separately: $(3x)^2 + 4^2 = 9x^2 + 16$ = wrong answer! This misses the crucial cross-multiplication term. Always remember the middle term $2ab = 2(3x)(4) = 24x$ in the perfect square formula.

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 can't I just square each term separately?

Squaring $(3x + 4)$ means multiplying the entire expression by itself, not just squaring individual terms. You get: $(3x + 4)(3x + 4)$ , which includes cross-terms when you distribute!

How do I remember the perfect square formula?

Think "First squared, twice the product, last squared": $(a + b)^2 = a^2 + 2ab + b^2$ . The middle term $2ab$ comes from multiplying each term by the other!

What if I have a negative sign like (3x - 4)²?

Use the same formula but be careful with signs! $(a - b)^2 = a^2 - 2ab + b^2$ . The middle term becomes negative because you're subtracting.

Can I use FOIL instead of the formula?

Absolutely! FOIL gives the same result: $(3x + 4)(3x + 4)$ = First + Outer + Inner + Last. The perfect square formula is just a shortcut for this special case.

How do I check if my expansion is correct?

Pick a simple number like $x = 1$ . Calculate $(3(1) + 4)^2 = 7^2 = 49$ , then check your expansion: $9(1)^2 + 24(1) + 16 = 9 + 24 + 16 = 49$ ✓

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