You have the equation:

\[
5y^2 + 8y - 3 = 0
\]

This is a quadratic equation in the standard form \( ay^2 + by + c = 0 \), where:

- \( a = 5 \)
- \( b = 8 \)
- \( c = -3 \)

We can solve this quadratic equation using the **quadratic formula**:

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

Let's substitute the values of \( a \), \( b \), and \( c \) into the formula:

\[
y = \frac{-8 \pm \sqrt{8^2 - 4(5)(-3)}}{2(5)}
\]
\[
y = \frac{-8 \pm \sqrt{64 + 60}}{10}
\]
\[
y = \frac{-8 \pm \sqrt{124}}{10}
\]
\[
y = \frac{-8 \pm 2\sqrt{31}}{10}
\]

So the solutions are:

\[
y = \frac{-8 + 2\sqrt{31}}{10} \quad \text{or} \quad y = \frac{-8 - 2\sqrt{31}}{10}
\]

Simplified:

\[
y = \frac{-4 + \sqrt{31}}{5} \quad \text{or} \quad y = \frac{-4 - \sqrt{31}}{5}
\]

These are the two solutions for \( y \).

Would you like further details on any step, or need clarification?

Here are 5 related questions to explore:
1. How do we derive the quadratic formula?
2. What happens if the discriminant (\( b^2 - 4ac \)) is negative?
3. How would we factor this quadratic equation, if possible?
4. How do we approximate the solutions for \( \sqrt{31} \)?
5. What are the properties of the graph of this quadratic equation?

**Tip:** Always check the discriminant (\( b^2 - 4ac \)) before applying the quadratic formula—it tells you if the roots are real, equal, or complex!

Learn how to solve the quadratic equation 5y^2 + 8y - 3 = 0 using the quadratic formula. This problem involves key algebraic concepts like the discriminant and is suitable for students in Grades 9-11. Get a detailed step-by-step solution and understand how the discriminant affects the nature of the roots.

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