Let's solve the equation step by step.

The given equation is:

\[
20.05y + 28y \cdot (1 + 1.083y) = 3.89 \cdot (1 + 1.083y)
\]

### Step 1: Expand and simplify
First, let's expand the terms on both sides.

#### Left Side:
Expand the term \( 28y \cdot (1 + 1.083y) \):
\[
20.05y + 28y \cdot 1 + 28y \cdot 1.083y
\]
\[
20.05y + 28y + 28 \cdot 1.083y^2
\]
\[
48.05y + 30.324y^2
\]

#### Right Side:
Expand the term \( 3.89 \cdot (1 + 1.083y) \):
\[
3.89 + 4.21347y
\]

So, the equation becomes:

\[
48.05y + 30.324y^2 = 3.89 + 4.21347y
\]

### Step 2: Rearrange the equation
Move all terms to one side to set the equation to zero:

\[
30.324y^2 + 48.05y - 4.21347y - 3.89 = 0
\]

Simplify further:

\[
30.324y^2 + 43.83653y - 3.89 = 0
\]

### Step 3: Solve the quadratic equation
This is the same quadratic equation we solved earlier:
\[
30.324y^2 + 43.83653y - 3.89 = 0
\]

Using the quadratic formula, the solutions for \( y \) are:

\[
y_1 \approx 0.0839 \quad \text{and} \quad y_2 \approx -1.5295
\]

So, the two possible solutions for \( y \) are:

1. \( y \approx 0.0839 \)
2. \( y \approx -1.5295 \)

Would you like further clarification or more details on this process?

### Related Questions:
1. Why is it important to expand and simplify terms in algebraic equations?
2. What role does the quadratic formula play in solving equations?
3. How can you determine if a quadratic equation has real or complex solutions?
4. What does each part of the quadratic formula represent?
5. How do you know when to use the quadratic formula?
6. Can a quadratic equation have only one solution?
7. What is the significance of the coefficients in a quadratic equation?
8. How would you interpret the solutions graphically?

**Tip:** Always double-check the expanded terms to ensure accuracy in your calculations, as errors can propagate and affect the final solution.

Learn how to solve the algebraic equation involving quadratic terms step-by-step. This problem covers key algebraic concepts such as quadratic equations and expansion techniques. Suitable for high school students.

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