Consider the expansion of (1 + x)
n
(1 − 2x)
4
, where n is a positive integer. If
the coefficient of x
2
is 54, find the coefficient of x.

We are asked to expand \( (1 + x)^n (1 - 2x)^4 \) and find the coefficient of \( x \), given that the coefficient of \( x^2 \) is 54.

### Step 1: Expand each factor using the binomial theorem
We use the binomial theorem to expand each factor separately.

#### Expansion of \( (1 + x)^n \):
\[
(1 + x)^n = 1 + \binom{n}{1}x + \binom{n}{2}x^2 + \dots
\]
The relevant terms for us are up to \( x^2 \), so we write:
\[
(1 + x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \dots
\]

#### Expansion of \( (1 - 2x)^4 \):
\[
(1 - 2x)^4 = 1 + \binom{4}{1}(-2x) + \binom{4}{2}(-2x)^2 + \dots
\]
The first few terms are:
\[
(1 - 2x)^4 = 1 - 8x + 24x^2 + \dots
\]

### Step 2: Multiply the two expansions
Now, we multiply the expansions \( (1 + x)^n \) and \( (1 - 2x)^4 \) to find the terms involving \( x \) and \( x^2 \).

#### Constant term (coefficient of \( x^0 \)):
\[
1 \times 1 = 1
\]

#### Coefficient of \( x \):
There are two ways to get the \( x \)-term:
1. \( 1 \times (-8x) = -8x \)
2. \( nx \times 1 = nx \)

Thus, the coefficient of \( x \) is:
\[
n - 8
\]

#### Coefficient of \( x^2 \):
There are three ways to get the \( x^2 \)-term:
1. \( 1 \times 24x^2 = 24x^2 \)
2. \( nx \times (-8x) = -8nx^2 \)
3. \( \frac{n(n-1)}{2}x^2 \times 1 = \frac{n(n-1)}{2}x^2 \)

Thus, the coefficient of \( x^2 \) is:
\[
24 - 8n + \frac{n(n-1)}{2}
\]
We are told that this coefficient is 54, so we set up the equation:
\[
24 - 8n + \frac{n(n-1)}{2} = 54
\]
Multiply through by 2 to eliminate the fraction:
\[
48 - 16n + n(n-1) = 108
\]
Simplifying the quadratic expression:
\[
48 - 16n + n^2 - n = 108
\]
\[
n^2 - 17n + 48 = 108
\]
\[
n^2 - 17n - 60 = 0
\]
Now solve the quadratic equation using the quadratic formula:
\[
n = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(1)(-60)}}{2(1)}
\]
\[
n = \frac{17 \pm \sqrt{289 + 240}}{2}
\]
\[
n = \frac{17 \pm \sqrt{529}}{2}
\]
\[
n = \frac{17 \pm 23}{2}
\]
Thus, \( n = \frac{17 + 23}{2} = 20 \) or \( n = \frac{17 - 23}{2} = -3 \).

Since \( n \) is a positive integer, we take \( n = 20 \).

### Step 3: Find the coefficient of \( x \)
The coefficient of \( x \) is \( n - 8 \). Substituting \( n = 20 \), we get:
\[
\text{Coefficient of } x = 20 - 8 = 12
\]

Thus, the coefficient of \( x \) is \( \boxed{12} \).

---

Would you like more details or clarifications? Here are some related questions:

1. How is the binomial theorem applied in expansions?
2. Why did we stop the expansion at \( x^2 \)?
3. Can this method be applied to other polynomials?
4. How do we solve quadratic equations in general?
5. How does the quadratic formula work in solving the equation?

**Tip:** When solving quadratic equations, always check if you can factorize it before using the quadratic formula!

Consider the expansion of (1 + x)^n (1 − 2x)^4, where n is a positive integer. If the coefficient of x^2 is 54, find the coefficient of x.

Explore the expansion of (1 + x)^n (1 − 2x)^4 and learn how to find the coefficient of x given the coefficient of x^2 is 54. This problem involves key concepts such as binomial expansion, polynomial multiplication, and solving quadratic equations. Suitable for advanced high school students in Grades 11-12.

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