https://file.aiquickdraw.com/mathbot/file/1730047947082c70uso51.jpg

The equation presented is:

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

To solve this quadratic equation without a calculator, we can use the **quadratic formula**:

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

Here:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -3 \)

### Step-by-Step Solution
1. **Substitute the values** into the formula:
   \[
   y = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2}
   \]
   
2. **Simplify** the expression:
   \[
   y = \frac{5 \pm \sqrt{25 + 24}}{4}
   \]

3. **Calculate** the discriminant:
   \[
   y = \frac{5 \pm \sqrt{49}}{4}
   \]

4. **Simplify the square root**:
   \[
   y = \frac{5 \pm 7}{4}
   \]

5. **Calculate the two possible solutions**:
   - **For the positive root**:
     \[
     y = \frac{5 + 7}{4} = \frac{12}{4} = 3
     \]
   
   - **For the negative root**:
     \[
     y = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}
     \]

### Final Answer
The solutions for \( y \) are:
\[
y = 3 \quad \text{and} \quad y = -\frac{1}{2}
\]

Would you like more details on any specific part? 

Here are some related questions that could expand your understanding:
1. How can we verify the solutions we found for \( y \) are correct?
2. What are the conditions for a quadratic equation to have real roots?
3. Can the quadratic formula be used for any type of quadratic equation?
4. How does the discriminant \( b^2 - 4ac \) determine the nature of the roots?
5. What are other methods to solve quadratic equations without using the quadratic formula?

**Tip:** When using the quadratic formula, always calculate the discriminant \( b^2 - 4ac \) first to check if the roots will be real or complex.

Solve for y in the equation 2y^2 - 5y - 3 = 0 without using a calculator.

Learn how to solve the quadratic equation 2y^2 - 5y - 3 with a detailed step-by-step solution. This problem explores algebraic concepts like the quadratic equation formula and uses the quadratic formula theorem. Suitable for high school students in Grades 9-12.

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