Let's solve both quadratic equations.

### 1. First equation: \( 5x^2 + 3x - 2 = 0 \)

This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 5 \)
- \( b = 3 \)
- \( c = -2 \)

We use the **quadratic formula**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values of \( a \), \( b \), and \( c \):
\[
x = \frac{-3 \pm \sqrt{3^2 - 4(5)(-2)}}{2(5)}
\]
\[
x = \frac{-3 \pm \sqrt{9 + 40}}{10}
\]
\[
x = \frac{-3 \pm \sqrt{49}}{10}
\]
\[
x = \frac{-3 \pm 7}{10}
\]

So the two solutions are:
\[
x_1 = \frac{-3 + 7}{10} = \frac{4}{10} = 0.4
\]
\[
x_2 = \frac{-3 - 7}{10} = \frac{-10}{10} = -1
\]

Thus, the solutions for \( x \) are:
\[
x = 0.4 \quad \text{or} \quad x = -1
\]

---

### 2. Second equation: \( 4y^2 - 7y + 3 = 0 \)

Here, the quadratic equation is in the form \( ay^2 + by + c = 0 \), where:
- \( a = 4 \)
- \( b = -7 \)
- \( c = 3 \)

Using the **quadratic formula**:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values of \( a \), \( b \), and \( c \):
\[
y = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(4)(3)}}{2(4)}
\]
\[
y = \frac{7 \pm \sqrt{49 - 48}}{8}
\]
\[
y = \frac{7 \pm \sqrt{1}}{8}
\]
\[
y = \frac{7 \pm 1}{8}
\]

So the two solutions are:
\[
y_1 = \frac{7 + 1}{8} = \frac{8}{8} = 1
\]
\[
y_2 = \frac{7 - 1}{8} = \frac{6}{8} = 0.75
\]

Thus, the solutions for \( y \) are:
\[
y = 1 \quad \text{or} \quad y = 0.75
\]

---

Do you want more details or have any questions?

Here are 5 related questions:
1. How can you verify if the solutions for \( x \) and \( y \) are correct?
2. What happens if the discriminant is negative in a quadratic equation?
3. Can you solve a quadratic equation by factoring instead of using the quadratic formula?
4. How does the quadratic formula relate to the graph of a quadratic function?
5. What are the conditions for a quadratic equation to have real or complex roots?

**Tip:** Always check the discriminant \( b^2 - 4ac \) before solving a quadratic equation, as it determines the nature of the roots.

Learn how to solve the quadratic equations 5x^2 + 3x - 2 and 4y^2 - 7y + 3 with detailed step-by-step solutions. Explore the quadratic formula and its application. Suitable for students in Grades 9-10.

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