Jika x=2p-4q dan y=-p+2q maka \frac{2x^2-3xy+y^2}{x^2-Y^2} berapa?

Given that \( x = 2p - 4q \) and \( y = -p + 2q \), we are asked to find the value of the expression:

\[
\frac{2x^2 - 3xy + y^2}{x^2 - y^2}.
\]

### Step 1: Express \( x^2 \), \( y^2 \), and \( xy \) in terms of \( p \) and \( q \).

First, calculate \( x^2 \), \( y^2 \), and \( xy \):

\[
x^2 = (2p - 4q)^2 = 4p^2 - 16pq + 16q^2,
\]

\[
y^2 = (-p + 2q)^2 = p^2 - 4pq + 4q^2,
\]

\[
xy = (2p - 4q)(-p + 2q).
\]

Now compute \( xy \):

\[
xy = 2p(-p + 2q) - 4q(-p + 2q) = -2p^2 + 4pq + 4pq - 8q^2,
\]

\[
xy = -2p^2 + 8pq - 8q^2.
\]

### Step 2: Compute \( 2x^2 - 3xy + y^2 \).

Next, calculate the numerator \( 2x^2 - 3xy + y^2 \):

\[
2x^2 = 2(4p^2 - 16pq + 16q^2) = 8p^2 - 32pq + 32q^2,
\]

\[
-3xy = -3(-2p^2 + 8pq - 8q^2) = 6p^2 - 24pq + 24q^2,
\]

\[
2x^2 - 3xy + y^2 = (8p^2 - 32pq + 32q^2) + (6p^2 - 24pq + 24q^2) + (p^2 - 4pq + 4q^2).
\]

Combine like terms:

\[
2x^2 - 3xy + y^2 = (8p^2 + 6p^2 + p^2) + (-32pq - 24pq - 4pq) + (32q^2 + 24q^2 + 4q^2),
\]

\[
2x^2 - 3xy + y^2 = 15p^2 - 60pq + 60q^2.
\]

### Step 3: Compute \( x^2 - y^2 \).

Calculate the denominator \( x^2 - y^2 \):

\[
x^2 - y^2 = (4p^2 - 16pq + 16q^2) - (p^2 - 4pq + 4q^2),
\]

\[
x^2 - y^2 = (4p^2 - p^2) + (-16pq + 4pq) + (16q^2 - 4q^2),
\]

\[
x^2 - y^2 = 3p^2 - 12pq + 12q^2.
\]

### Step 4: Simplify the expression.

Now simplify the expression:

\[
\frac{2x^2 - 3xy + y^2}{x^2 - y^2} = \frac{15p^2 - 60pq + 60q^2}{3p^2 - 12pq + 12q^2}.
\]

Factor both the numerator and the denominator:

\[
\frac{15(p^2 - 4pq + 4q^2)}{3(p^2 - 4pq + 4q^2)}.
\]

This simplifies to:

\[
\frac{15}{3} = 5.
\]

### Final Answer

The value of the expression is:

\[
\frac{2x^2 - 3xy + y^2}{x^2 - y^2} = 5.
\]

Would you like more details, or do you have any other questions?

Here are five related questions you might find interesting:
1. What if the expression was \( \frac{2x^2 + 3xy + y^2}{x^2 + y^2} \)?
2. How do you solve equations with multiple variables, such as \( x = 2p - 4q \) and \( y = -p + 2q \)?
3. How do you find the discriminant of a quadratic equation?
4. What is the geometric interpretation of expressions like \( x^2 - y^2 \)?
5. How can you simplify expressions involving powers and cross terms in algebra?

**Tip:** When simplifying algebraic expressions, always look for common factors in the numerator and denominator to reduce the expression efficiently.

Learn how to calculate the expression (2x^2 - 3xy + y^2) / (x^2 - y^2) using algebraic techniques. Understand the steps involved in simplifying the expression with detailed solutions. Suitable for advanced high school students.

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