We are tasked with solving the equation:

\[
\frac{(2 + 2x)^2}{(2 - x)(3 - x)} = 1
\]

### Step-by-step solution:

1. **Expand the numerator**: 
   \[
   (2 + 2x)^2 = (2 + 2x)(2 + 2x) = 4 + 8x + 4x^2
   \]
   So the equation becomes:
   \[
   \frac{4x^2 + 8x + 4}{(2 - x)(3 - x)} = 1
   \]

2. **Multiply both sides by \((2 - x)(3 - x)\)** to eliminate the denominator on the left-hand side:
   \[
   4x^2 + 8x + 4 = (2 - x)(3 - x)
   \]

3. **Expand the right-hand side**:
   \[
   (2 - x)(3 - x) = 6 - 3x - 2x + x^2 = x^2 - 5x + 6
   \]
   So now the equation is:
   \[
   4x^2 + 8x + 4 = x^2 - 5x + 6
   \]

4. **Move all terms to one side**:
   \[
   4x^2 + 8x + 4 - x^2 + 5x - 6 = 0
   \]
   Simplify:
   \[
   3x^2 + 13x - 2 = 0
   \]

5. **Solve the quadratic equation**: 
   The quadratic equation is:
   \[
   3x^2 + 13x - 2 = 0
   \]
   Use the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \(a = 3\), \(b = 13\), and \(c = -2\). 

   Calculate the discriminant:
   \[
   \Delta = b^2 - 4ac = 13^2 - 4(3)(-2) = 169 + 24 = 193
   \]
   Thus:
   \[
   x = \frac{-13 \pm \sqrt{193}}{6}
   \]

6. **Final solution**:
   \[
   x = \frac{-13 + \sqrt{193}}{6} \quad \text{or} \quad x = \frac{-13 - \sqrt{193}}{6}
   \]

These are the two solutions.

Would you like more details on any of the steps?

### 5 Related Questions:
1. What are the properties of quadratic equations?
2. How do you find the discriminant of a quadratic equation?
3. Can the quadratic formula be applied to any quadratic equation?
4. What is the process of expanding binomials like \((2 + 2x)^2\)?
5. How does multiplying by the denominator help simplify an equation?

### Tip:
Always simplify both sides of an equation as much as possible before applying advanced methods like the quadratic formula.

This page provides a detailed step-by-step solution for the equation (2 + 2x)^2 / (2 - x)(3 - x) = 1. The solution covers algebraic concepts, including binomial expansion, rational expressions, and solving quadratic equations using the quadratic formula. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation