The equation you need to solve is:

$x^2 + 9x + 18 = 0$

We can solve this quadratic equation using the quadratic formula:

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

For the equation $x^2 + 9x + 18 = 0$, the coefficients are:

• $a = 1$
• $b = 9$
• $c = 18$

Now, let's substitute these values into the quadratic formula:

$x = \frac{-9 \pm \sqrt{9^2 - 4(1)(18)}}{2(1)}$

First, calculate the discriminant:

$\Delta = 9^2 - 4(1)(18) = 81 - 72 = 9$

Now, calculate the two possible values for $x$:

$x_1 = \frac{-9 + \sqrt{9}}{2} = \frac{-9 + 3}{2} = \frac{-6}{2} = -3$

$x_2 = \frac{-9 - \sqrt{9}}{2} = \frac{-9 - 3}{2} = \frac{-12}{2} = -6$

So, the two solutions are:

• $x_1 = -3$ (the larger solution)
• $x_2 = -6$ (the smaller solution)

Therefore:

• Box 1 (Larger Solution) = $-3$
• Box 2 (Smaller Solution) = $-6$

Would you like further clarification on the steps?

Here are five questions related to this:

1. How is the discriminant used to determine the nature of the roots in a quadratic equation?
2. Can a quadratic equation have complex solutions? If so, when?
3. What is the difference between solving quadratics by factoring versus using the quadratic formula?
4. How does changing the sign of the $b$-term affect the solutions?
5. How does the quadratic formula handle coefficients that are fractions or irrational numbers?

Tip: Always check the discriminant ($\Delta$) to understand the nature of the roots—whether they're real, equal, or complex.