The given quadratic equation is:

$x^2 + x - 18 = 0$

To solve it, we can use the quadratic formula, which is:

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

For the equation $x^2 + x - 18 = 0$, we have:

• $a = 1$ (coefficient of $x^2$)
• $b = 1$ (coefficient of $x$)
• $c = -18$ (constant term)

Step 1: Calculate the discriminant

$\Delta = b^2 - 4ac = 1^2 - 4(1)(-18) = 1 + 72 = 73$

Step 2: Apply the quadratic formula

$x = \frac{-1 \pm \sqrt{73}}{2(1)} = \frac{-1 \pm \sqrt{73}}{2}$

Step 3: Find the two solutions

$x_1 = \frac{-1 + \sqrt{73}}{2}, \quad x_2 = \frac{-1 - \sqrt{73}}{2}$

The solutions are:

$x_1 \approx \frac{-1 + 8.54}{2} = \frac{7.54}{2} = 3.77$ $x_2 \approx \frac{-1 - 8.54}{2} = \frac{-9.54}{2} = -4.77$

Thus, the solutions are approximately: $x_1 \approx 3.77, \quad x_2 \approx -4.77$

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to further your understanding:

1. How do we know when to use the quadratic formula?
2. Can we factor this quadratic equation directly instead of using the formula?
3. What does the discriminant tell us about the nature of the solutions?
4. How would the solutions change if the discriminant were negative?
5. Can you solve a quadratic equation by completing the square?

Tip: The discriminant ($b^2 - 4ac$) helps you determine if a quadratic equation has real, complex, or equal roots!