Expand (4b-3)(4b-3): Converting to Perfect Square Form

Binomial Squaring with Perfect Square Trinomials

(4b−3)(4b−3) (4b-3)(4b-3)

Rewrite the above expression as an exponential summation expression:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Express the expression as a power expression and as a sum
00:03 A factor multiplied by itself is essentially squared
00:13 Therefore the parentheses are squared
00:19 We'll use the shortened multiplication formulas to expand the parentheses
00:38 Let's calculate the squares and products
00:56 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

(4b−3)(4b−3) (4b-3)(4b-3)

Rewrite the above expression as an exponential summation expression:

2

Step-by-step solution

To solve this problem, we will apply the square of a binomial formula.

The given expression is (4b−3)(4b−3)(4b-3)(4b-3). We recognize this as the square of a binomial, which can be rewritten as (4b−3)2(4b-3)^2. To expand this expression, we use the formula:

(a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2

In our expression, a=4ba = 4b and b=3b = 3. Let's apply the formula:

  • Calculate a2a^2:
    a2=(4b)2=16b2a^2 = (4b)^2 = 16b^2
  • Calculate −2ab-2ab:
    −2ab=−2(4b)(3)=−24b-2ab = -2(4b)(3) = -24b
  • Calculate b2b^2:
    b2=(3)2=9b^2 = (3)^2 = 9

Putting it all together, we have:

(4b−3)2=16b2−24b+9(4b-3)^2 = 16b^2 - 24b + 9

Therefore, the exponential summation expression is (4b−3)2(4b-3)^2, with the expanded form:

16b2−24b+916b^2 - 24b + 9

This matches choice 3, confirming our solution.

3

Final Answer

(4b−3)2 (4b-3)^2

16b2−24b+9 16b^2-24b+9

Key Points to Remember

Essential concepts to master this topic
  • Pattern Recognition: (a−b)(a−b)=(a−b)2 (a-b)(a-b) = (a-b)^2 identifies repeated multiplication
  • Formula Application: (a−b)2=a2−2ab+b2 (a-b)^2 = a^2 - 2ab + b^2 where a=4b, b=3
  • Verification: Check middle term sign: −2(4b)(3)=−24b -2(4b)(3) = -24b confirms negative ✓

Common Mistakes

Avoid these frequent errors
  • Getting positive middle term in expansion
    Don't write (4b−3)2=16b2+24b+9 (4b-3)^2 = 16b^2 + 24b + 9 = wrong sign! This happens when students forget the negative in -2ab. Always remember (a−b)2=a2−2ab+b2 (a-b)^2 = a^2 - 2ab + b^2 has a negative middle term.

Practice Quiz

Test your knowledge with interactive questions

\( (4b-3)(4b-3) \)

Rewrite the above expression as an exponential summation expression:

FAQ

Everything you need to know about this question

Why does (4b−3)(4b−3) (4b-3)(4b-3) become (4b−3)2 (4b-3)^2 ?

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When you multiply something by itself, that's the definition of squaring! Just like 5×5=52 5 \times 5 = 5^2 , we have (4b−3)×(4b−3)=(4b−3)2 (4b-3) \times (4b-3) = (4b-3)^2 .

How do I remember the perfect square trinomial formula?

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Think "First squared, minus twice the product, plus last squared": (a−b)2=a2−2ab+b2 (a-b)^2 = a^2 - 2ab + b^2 . The middle term is always negative when you have (a-b)²!

What if I forget the formula and just use FOIL?

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FOIL works perfectly! (4b−3)(4b−3) (4b-3)(4b-3) gives: First: 16b2 16b^2 , Outer: −12b -12b , Inner: −12b -12b , Last: 9 9 . Combine like terms: 16b2−24b+9 16b^2 - 24b + 9 .

Why is the middle term −24b -24b and not −12b -12b ?

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When you FOIL, you get two middle terms: −12b -12b and −12b -12b . Adding them gives −12b+(−12b)=−24b -12b + (-12b) = -24b . That's why the formula has -2ab!

How can I check if my expansion is correct?

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Pick a simple value like b=1 b = 1 . Then (4(1)−3)2=12=1 (4(1)-3)^2 = 1^2 = 1 and 16(1)2−24(1)+9=16−24+9=1 16(1)^2 - 24(1) + 9 = 16 - 24 + 9 = 1 . Both equal 1, so it's correct!

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