Perfect Square Expansion: Complete (2x-?)² = 4x²-12x+?

Perfect Square Binomials with Missing Terms

Fill in the blanks:

(2x−?)2=4x2−12x+? (2x-?)^2=4x^2-12x+\text{?}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:08 Let's fill in the missing parts together.
00:11 Use the simple multiplication rules to expand the parentheses.
00:15 We'll call the unknown number A.
00:22 Now, calculate A squared and multiply it out.
00:34 Match the equal parts and solve for A.
00:43 Make A the focus by isolating it.
00:55 This is our solution for A.
00:59 Now plug the value of A into the right spots.
01:03 Calculate three squared and replace.
01:07 Great job! That's the answer to our question.

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Fill in the blanks:

(2x−?)2=4x2−12x+? (2x-?)^2=4x^2-12x+\text{?}

2

Step-by-step solution

To solve the problem, we'll follow these steps:

  • Step 1: Identify the given form and match it with the square of a difference formula
  • Step 2: Determine the values of 'a' and 'b' such that the expanded form matches both sides of the equation
  • Step 3: Calculate the missing value in the expression

Now, let's work through each step:
Step 1: We are given the expression (2x−?)2=4x2−12x+? (2x-?)^2=4x^2-12x+\text{?} .
Step 2: Using the standard formula for a perfect square expansion:
(2x−b)2=(2x)2−2⋅2x⋅b+b2=4x2−4xb+b2(2x-b)^2 = (2x)^2 - 2 \cdot 2x \cdot b + b^2 = 4x^2 - 4xb + b^2.
By matching coefficients, in 4x2−12x+?4x^2 - 12x + \text{?}, we see 4xb=12x4xb = 12x. Thus, b=3b = 3.
Step 3: Substitute b=3b = 3 into b2b^2 to get the constant term: b2=32=9b^2 = 3^2 = 9.

Therefore, the solution to the problem is 3, 93,\text{ }9.

3

Final Answer

3, 9 3,\text{ }9

Key Points to Remember

Essential concepts to master this topic
  • Formula: (a−b)2=a2−2ab+b2 (a-b)^2 = a^2 - 2ab + b^2 expansion pattern
  • Technique: Match coefficients: −4xb=−12x -4xb = -12x , so b=3 b = 3
  • Check: Expand (2x−3)2=4x2−12x+9 (2x-3)^2 = 4x^2 - 12x + 9 ✓

Common Mistakes

Avoid these frequent errors
  • Finding the missing constant by dividing the middle term coefficient
    Don't divide -12 by 4 to get -3 for the constant term = wrong answer! This ignores that the constant is b², not b. Always find b first from the middle term, then square it: b = 3, so b² = 9.

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

How do I know which number goes in the first blank?

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Look at the middle term coefficient! In 4x2−12x 4x^2 - 12x , we have -12x. Since the formula gives us -2ab = -2(2x)(b) = -4xb, we need -4xb = -12x, so b = 3.

Why can't I just guess and check the answer choices?

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While that might work here, understanding the pattern helps you solve any perfect square problem! Plus, you'll make fewer mistakes when you know why the answer is correct.

What if the first term wasn't 4x²?

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The same process works! If you had (3x−?)2 (3x - ?)^2 , you'd get 9x2−6xb+b2 9x^2 - 6xb + b^2 . Always match the middle term coefficient to find the missing value.

How do I remember the perfect square formula?

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Think FOIL backwards! (a−b)2 (a-b)^2 means multiply (a−b)(a−b) (a-b)(a-b) :

  • First terms: a·a = a²
  • Outer + Inner: a·(-b) + (-b)·a = -2ab
  • Last terms: (-b)·(-b) = b²

Can this method work for addition too?

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Absolutely! For (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2 , the process is identical. Just watch the signs - addition gives a positive middle term, subtraction gives negative.

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