The image contains a quadratic equation $2x^2 + 15x + 18 = 0$ and the corresponding solutions for $x$. However, the solutions shown do not appear to be correct given the equation.

Let's solve this quadratic equation step by step using the quadratic formula:

Quadratic Formula:

For a quadratic equation in the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case:

• $a = 2$
• $b = 15$
• $c = 18$

Step 1: Calculate the discriminant

The discriminant is: $\Delta = b^2 - 4ac = 15^2 - 4(2)(18) = 225 - 144 = 81$

Step 2: Find the roots

Now, apply the quadratic formula: $x = \frac{-15 \pm \sqrt{81}}{2(2)} = \frac{-15 \pm 9}{4}$

This gives two solutions:

1. $x_1 = \frac{-15 + 9}{4} = \frac{-6}{4} = -\frac{3}{2}$
2. $x_2 = \frac{-15 - 9}{4} = \frac{-24}{4} = -6$

Final Solutions:

The correct solutions are: $x = -\frac{3}{2} \quad \text{or} \quad x = -6$

Would you like further details or have any questions about this solution?

Here are 5 related questions to explore:

1. What is the significance of the discriminant in the quadratic formula?
2. How would you factor this quadratic equation if solving without the quadratic formula?
3. Can you derive the quadratic formula from completing the square?
4. What happens if the discriminant is negative or zero?
5. How would the solutions change if the constant term (18) were altered?

Tip: Always check the discriminant value to determine the nature of the roots (real or complex).