Solving a quadratic equation using the polynomial solver on a Casio calculator.

Let's solve the quadratic equation step by step and verify the given information.

We are given:

$$
x^2 + 11x + 18 = 0
$$

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Step 1: Solve using factoring

We want two numbers that multiply to 18 (the constant term) and add up to 11 (the coefficient of $x$).

Let’s find such numbers:

- Factors of 18:
$1 \times 18$, $2 \times 9$, $3 \times 6$

Check which pair adds to 11:
- $2 + 9 = 11$ ✔

So we can factor:

$$
x^2 + 11x + 18 = (x + 2)(x + 9) = 0
$$

Set each factor equal to zero:

- $x + 2 = 0$ → $x = -2$
- $x + 9 = 0$ → $x = -9$

✔ So, the solutions are:
$$
\boxed{x = -2} \quad \text{or} \quad \boxed{x = -9}
$$

This matches what is shown in the image.

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Step 2: Find the vertex (minimum point)

For a quadratic in the form $y = ax^2 + bx + c$, the vertex (minimum or maximum point) occurs at:

$$
x = -\frac{b}{2a}
$$

Here, $a = 1$, $b = 11$, so:

$$
x = -\frac{11}{2(1)} = -5.5
$$

Now plug $x = -5.5$ into the original equation to find the $y$-coordinate:

$$
y = (-5.5)^2 + 11(-5.5) + 18
= 30.25 - 60.5 + 18
= -12.25
$$

So the minimum point is:

$$
\boxed{(-5.5, -12.25)}
$$

This matches the calculator display showing "Min of y=ax²+bx+c" with $y = -12.25$. Since $a > 0$, the parabola opens upward, so this is indeed a minimum.

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Step 3: Complete the square

Start with:

$$
x^2 + 11x + 18 = 0
$$

To complete the square:

1. Move constant to other side:
$$
x^2 + 11x = -18
$$

2. Take half of 11: $\frac{11}{2}$, square it: $\left(\frac{11}{2}\right)^2 = \frac{121}{4}$

3. Add to both sides:
$$
x^2 + 11x + \frac{121}{4} = -18 + \frac{121}{4}
$$

4. Simplify right-hand side:
$$
-18 = -\frac{72}{4}, \quad -\frac{72}{4} + \frac{121}{4} = \frac{49}{4}
$$

So:
$$
\left(x + \frac{11}{2}\right)^2 = \frac{49}{4}
$$

Now write as:
$$
\left(x + \frac{11}{2}\right)^2 - \frac{49}{4} = 0
$$

✔ This matches the "Completed Square" form shown in the image.

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Final Summary

- Solutions: $x = -2$ or $x = -9$
- Minimum Point: $(-5.5, -12.25)$
- Completed Square Form: $(x + \frac{11}{2})^2 - \frac{49}{4} = 0$

All values in the image are correct.

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Calculator Use (Polynomial Solver)

On a Casio ClassWiz calculator:
1. Press `MENU` → Select `EQN` (Equation)
2. Choose `2` for quadratic equation
3. Input coefficients: $a = 1$, $b = 11$, $c = 18$
4. Press `EXE` → Calculator gives roots: $x = -2$, $x = -9$
5. To find minimum: use `GRAPH` mode or analyze vertex from formula

The displayed value `-12.25` on the screen confirms the minimum y-value, which aligns with our calculation.

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✔ Conclusion: The problem is correctly solved. All steps — factoring, vertex, completing the square — are accurate.