Bit (a lot) of trouble solving this: The landing for the steps leading up to a county courthouse is shaped like a trapezoid. The area of the landing is 1500 square feet. The shorter base of the trapezoid is 15 feet longer than the height. The longer base is 5 feet longer than 3 times the longer base? (Btw Solve By Completing the Square)

S = H + 15

L = 3S + 5 = 3H + 50

A = H (S + L) / 2 = H (4H + 65) / 2

A = 2 H^2 + 32.5 H

To solve this problem, let's assign variables to the different components of the trapezoid.

Let:
H = height of the trapezoid
B1 = length of the shorter base
B2 = length of the longer base

From the given information, we can form the following equations:

1. The area of the landing is 1500 square feet:
Area = (B1 + B2) * H / 2
1500 = (B1 + B2) * H / 2

2. The shorter base is 15 feet longer than the height:
B1 = H + 15

3. The longer base is 5 feet longer than 3 times the longer base:
B2 = 3B1 + 5

Now, let's substitute the values from equations 2 and 3 into equation 1 and solve using completing the square:

1500 = [(H + 15) + (3(H + 15) + 5)] * H / 2
3000 = [2H + 15 + 6H + 45 + 2H] * H
3000 = (10H + 60) * H
3000 = 10H^2 + 60H
10H^2 + 60H - 3000 = 0

Next, divide the entire equation by 10 to simplify it:
H^2 + 6H - 300 = 0

To complete the square, add half of the coefficient of H squared to both sides of the equation:
H^2 + 6H + (6/2)^2 - (6/2)^2 - 300 = 0
H^2 + 6H + 9 - 9 - 300 = 0
H^2 + 6H + 9 - 309 = 0
(H + 3)^2 - 309 = 0

Rearrange the equation:
(H + 3)^2 = 309

Take the square root of both sides:
√[(H + 3)^2] = ±√309
H + 3 = ±√309

Now, solve for H by subtracting 3 from both sides of the equation:
H = -3 ± √309

Finally, substitute the value of H into equations 2 and 3 to find B1 and B2:

B1 = H + 15
B1 = (-3 ± √309) + 15

B2 = 3B1 + 5
B2 = 3[(-3 ± √309) + 15] + 5

These are the expressions for the values of B1 and B2, which can be further simplified if desired.

To solve this problem using completing the square, let's break down the information given.

We are given that the area of the landing (A) is 1500 square feet. The formula for finding the area of a trapezoid is A = (1/2) × (b1 + b2) × h, where b1 and b2 are the bases and h is the height.

So, we can write the equation for the area of the landing as 1500 = (1/2) × (b1 + b2) × h.

Next, we know that the shorter base (b2) is 15 feet longer than the height (h). So we can write b2 = h + 15.

Finally, we are given that the longer base (b1) is 5 feet longer than 3 times the shorter base (b2). So we can write b1 = 3b2 + 5.

Now, substituting the values of b1 and b2 into the equation for the area, we have 1500 = (1/2) × (3b2 + 5 + b2) × h.

Simplifying, we get 1500 = (1/2) × (4b2 + 5) × h.

Now, let's solve this equation by completing the square.

First, divide both sides by (1/2) to get rid of the fraction:
3000 = (4b2 + 5) × h.

Next, let's separate the variables and constants:
3000 = 4b2h + 5h.

Now, we want to complete the square for the variable term 4b2h. To do this, we need to take half of the coefficient of h, square it, and add it to both sides of the equation. The coefficient of h is 5, so (1/2) × 5 = 2.5. Squaring 2.5 gives us 6.25.

Adding 6.25 to both sides of the equation, we get:
3006.25 = 4b2h + 5h + 6.25.

Now, we can rewrite the left side as a perfect square:
3006.25 = (2b2 + 2.5h)^2.

Taking the square root of both sides, we have:
√3006.25 = 2b2 + 2.5h.

Simplifying the square root, we get:
√3006.25 = 2b2 + 2.5h.

Now, to solve for the values of b2 and h, you can substitute different values for the square root of 3006.25 on the right side and solve the equation.