The given equation is:

\[
2x^2 + 11x + 5 = 0
\]

To solve this quadratic equation, we can use the **quadratic formula**:

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

Where:
- \( a = 2 \) (the coefficient of \( x^2 \)),
- \( b = 11 \) (the coefficient of \( x \)),
- \( c = 5 \) (the constant term).

### Step 1: Calculate the discriminant (\( \Delta \))
The discriminant is:

\[
\Delta = b^2 - 4ac = 11^2 - 4(2)(5) = 121 - 40 = 81
\]

### Step 2: Apply the quadratic formula
Now substitute the values into the quadratic formula:

\[
x = \frac{-11 \pm \sqrt{81}}{2(2)} = \frac{-11 \pm 9}{4}
\]

### Step 3: Solve for the two roots
Now, solve for the two possible values of \( x \):

1. \( x_1 = \frac{-11 + 9}{4} = \frac{-2}{4} = -\frac{1}{2} \)
2. \( x_2 = \frac{-11 - 9}{4} = \frac{-20}{4} = -5 \)

### Final Answer:
The solutions to the equation are:

\[
x = -\frac{1}{2} \quad \text{or} \quad x = -5
\]

Would you like more details or have any questions?

Here are some related questions you can explore:

1. How do you solve quadratic equations by factoring?
2. What is the importance of the discriminant in quadratic equations?
3. Can quadratic equations always be solved using the quadratic formula?
4. How do complex solutions arise in quadratic equations?
5. How does the graph of \( 2x^2 + 11x + 5 = 0 \) look like?

**Tip:** The discriminant helps determine the number and nature of solutions: positive means two real solutions, zero means one real solution, and negative means two complex solutions.

Explore how to solve the quadratic equation 2x^2 + 11x + 5 using the quadratic formula. This detailed step-by-step solution involves calculating the discriminant and applying the formula, perfect for Grades 8-10 students.

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