Expand [4(2-x)]²: Square Brackets and Binomial Expression

Q: Why can't I just square the 4 and square the (2-x) separately?

Because of the order of operations! The brackets mean you must treat $4(2-x)$ as one unit. You need to distribute first, then square the entire result.

Q: What's the difference between (a-b)² and (a+b)²?

The middle term changes sign! $(a-b)^2 = a^2 - 2ab + b^2$ but $(a+b)^2 = a^2 + 2ab + b^2$. Notice the minus sign in the first formula.

Q: How do I remember the binomial square formula?

Think "First, Twice, Last": First term squared, twice the product of both terms, last term squared. For $(8-4x)^2$: 8² - 2(8)(4x) + (4x)²

Q: Why is my final answer in a different order than the given options?

Standard form for polynomials puts the highest power first. So $64 - 64x + 16x^2$ should be rewritten as $16x^2 - 64x + 64$.

Q: Can I check my answer by substituting a value?

Yes! Try x = 0: Original gives $[4(2-0)]^2 = 64$, and your answer $16(0)^2 - 64(0) + 64 = 64$ ✓

Binomial Expansion with Algebraic Distribution

$\lbrack4(2-x)\rbrack^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 Open brackets properly, multiply by each factor

00:18 Use the shortened multiplication formulas to open the brackets

00:26 In this case, 8 is A

00:30 4X is B

00:38 Calculate 8 squared

00:42 Solve the multiplications

00:49 Raise each factor in the multiplication to the second power

00:57 Use the commutative law and arrange the exercise

01:03 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

$\lbrack4(2-x)\rbrack^2=$

Step-by-step solution

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

Step 1: Simplify inside the bracket.
Step 2: Apply the square of a binomial formula.
Step 3: Expand and simplify the expression.

Now, let's work through each step:

Step 1: Simplify inside the bracket:
The expression inside the bracket is $4(2-x)$ . Multiplying through by 4, we have:
$= 4 \times (2-x) = 8 - 4x$ .

Step 2: Apply the square of a binomial formula:
We want to compute $(8 - 4x)^2$ . According to the binomial formula:
$(a-b)^2 = a^2 - 2ab + b^2$ .

Let $a = 8$ and $b = 4x$ . Substituting these into the formula gives us:
$(8 - 4x)^2 = 8^2 - 2(8)(4x) + (4x)^2$ .

Step 3: Expand and simplify the expression:
$= 64 - 64x + 16x^2$ .

The correctly expanded expression is:
Therefore, the solution to the problem is $\mathbf{16x^2 - 64x + 64}$ .

Final Answer

$16x^2-64x+64$

Key Points to Remember

Essential concepts to master this topic

Distribution First: Multiply 4 through the parentheses before squaring
Binomial Formula: Use (a-b)² = a² - 2ab + b² for (8-4x)²
Check: Expand step by step: 64 - 64x + 16x² ✓

Common Mistakes

Avoid these frequent errors

Squaring without distributing first
Don't square [4(2-x)]² as 16(2-x)² = 16(4-4x+x²) = wrong answer! This skips the crucial distribution step and creates calculation errors. Always distribute 4(2-x) = 8-4x first, then apply the binomial square formula to (8-4x)².

Practice Quiz

Test your knowledge with interactive questions

Declares the given expression as a sum

$$ (7b-3x)^2 $$

FAQ

Everything you need to know about this question

Why can't I just square the 4 and square the (2-x) separately?

Because of the order of operations! The brackets mean you must treat $4(2-x)$ as one unit. You need to distribute first, then square the entire result.

What's the difference between (a-b)² and (a+b)²?

The middle term changes sign! $(a-b)^2 = a^2 - 2ab + b^2$ but $(a+b)^2 = a^2 + 2ab + b^2$ . Notice the minus sign in the first formula.

How do I remember the binomial square formula?

Think "First, Twice, Last": First term squared, twice the product of both terms, last term squared. For $(8-4x)^2$ : 8² - 2(8)(4x) + (4x)²

Why is my final answer in a different order than the given options?

Standard form for polynomials puts the highest power first. So $64 - 64x + 16x^2$ should be rewritten as $16x^2 - 64x + 64$ .

Can I check my answer by substituting a value?

Yes! Try x = 0: Original gives $[4(2-0)]^2 = 64$ , and your answer $16(0)^2 - 64(0) + 64 = 64$ ✓

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