We have the quadratic equation:

\[
0.2x^2 + 3x - 20 = 0
\]

To solve this equation, we can either use the quadratic formula or try factoring. Since factoring may not be straightforward due to the decimal coefficient, let's proceed with the **quadratic formula**:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this case, the coefficients are:

- \( a = 0.2 \)
- \( b = 3 \)
- \( c = -20 \)

### Step 1: Calculate the discriminant
\[
\text{Discriminant} = b^2 - 4ac = 3^2 - 4(0.2)(-20)
\]
\[
\text{Discriminant} = 9 - 4(0.2)(-20) = 9 + 16 = 25
\]

### Step 2: Apply the quadratic formula
\[
x = \frac{-3 \pm \sqrt{25}}{2(0.2)}
\]
\[
x = \frac{-3 \pm 5}{0.4}
\]

### Step 3: Solve for \( x \)
1. For the positive root:
\[
x = \frac{-3 + 5}{0.4} = \frac{2}{0.4} = 5
\]

2. For the negative root:
\[
x = \frac{-3 - 5}{0.4} = \frac{-8}{0.4} = -20
\]

Thus, the two solutions are:

\[
x = 5 \quad \text{and} \quad x = -20
\]

Would you like any further details or clarification?

### Related Questions:
1. How does the discriminant help in determining the nature of the solutions?
2. Can you explain how to solve this equation by factoring instead of using the quadratic formula?
3. How can you verify the solutions for the equation by substituting them back?
4. What does it mean when the discriminant is negative?
5. How would the solutions change if the coefficient of \(x^2\) was different?

### Tip:
Always check the discriminant before applying the quadratic formula—it tells you whether you'll get real or complex solutions!

Learn how to solve the quadratic equation 0.2x² + 3x - 20 = 0 with a detailed step-by-step explanation using the quadratic formula. This solution involves calculating the discriminant and finding both real roots. Suitable for students in Grades 8-10.

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