We are given a triangular pennant, with the following details:

• The base is $z$ inches long.
• The height is $2z + 6$ inches, which is 6 inches longer than twice the base.
• The area $A$ of the triangle is 180 square inches.

We need to find the height.

Formula:

The area of a triangle is given by the formula: $A = \frac{1}{2} \times \text{base} \times \text{height}$

Substituting the given values:

$180 = \frac{1}{2} \times z \times (2z + 6)$

Solve for $z$:

1. Multiply both sides of the equation by 2 to eliminate the fraction: $360 = z \times (2z + 6)$

2. Distribute $z$: $360 = 2z^2 + 6z$

3. Rearrange the equation into standard quadratic form: $2z^2 + 6z - 360 = 0$

4. Divide the entire equation by 2 to simplify: $z^2 + 3z - 180 = 0$

Solve the quadratic equation using the quadratic formula:

The quadratic formula is: $z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $z^2 + 3z - 180 = 0$, we have $a = 1$, $b = 3$, and $c = -180$.

Now, let's calculate the value of $z$.The solutions for $z$ are $z = 12$ and $z = -15$. Since the base of the triangle cannot be negative, we select $z = 12$.

Find the height:

The height of the pennant is $2z + 6$, so: $\text{Height} = 2(12) + 6 = 24 + 6 = 30 \text{ inches}.$

Thus, the height of the pennant is 30 inches.

Would you like more details on the process or have any questions?

Here are 5 related questions to expand on this problem:

1. What is the perimeter of the triangular pennant if the third side is unknown?
2. How would the area change if the base was increased by 5 inches?
3. Can you verify the solution by plugging $z = 12$ into the area formula again?
4. How do we derive the quadratic formula from completing the square?
5. What is the geometric significance of a negative solution in this context?

Tip: When solving quadratic equations, always verify if both solutions are feasible in the context of the problem.