A contractor leans a 23-foot ladder against a building. The distance from the ground to the top of the ladder is 9 feet more than the distance from the building to the base of the ladder. How far up the building is the ladder to the nearest tenth of a foot?

x^2 + (x-9)^2 = 23^2

2 x^2 - 18 x + 81 = 529
2 x^2 - 18 x - 448 = 0
x^2 - 9 x - 224 = 0
well 20 * 11 = 220 so about 20 but use
https://www.mathsisfun.com/quadratic-equation-solver.html

Well, let's see here. If the distance from the ground to the top of the ladder is 9 feet more than the distance from the building to the base of the ladder, then I guess you could say the ladder is feeling a bit top-heavy, huh?

But fear not, my friend! We can solve this! Let's call the distance from the building to the base of the ladder "x". That means the distance from the ground to the top of the ladder is "x + 9".

Now, we have a lovely mathematical relationship between the height of the ladder and the base. The ladder, being a graceful creature, forms a right angle with the ground and building. So, we can use the Pythagorean theorem to find the height!

According to the Pythagorean theorem, the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. In this case, the height of the ladder is the hypotenuse.

So, x^2 + (x + 9)^2 = 23^2. Now we can solve for x!

x^2 + (x^2 + 18x + 81) = 529
2x^2 + 18x +81 = 529
2x^2 + 18x - 448 = 0

Using some fancy math magic, we find that x is approximately 8.6 feet.

So, my dear friend, the ladder is about 8.6 feet up the building! Give or take a few clown shoes.

Let's denote the distance from the ground to the base of the ladder as "x" feet. According to the problem, the distance from the ground to the top of the ladder is 9 feet more than the distance from the building to the base of the ladder.

This means the distance from the ground to the top of the ladder is x + 9 feet.

We can use the Pythagorean theorem to solve for the distance up the building. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this scenario, the ladder acts as the hypotenuse, and the distance from the base of the ladder to the top of the building as well as the distance from the ground to the base of the ladder form the other two sides of the triangle.

Using the Pythagorean theorem, we can write:
(x + 9)^2 = x^2 + 23^2

Expanding the equation:
x^2 + 18x + 81 = x^2 + 529

Simplifying and rearranging the equation:
18x = 448

Dividing both sides by 18:
x = 24.89

Therefore, the distance up the building, to the nearest tenth of a foot, is approximately 24.9 feet.

To solve this problem, we can use the Pythagorean theorem, which states that the square of the hypotenuse (in this case, the length of the ladder) is equal to the sum of the squares of the other two sides (the distance from the ground to the base of the ladder and the distance from the building to the base of the ladder).

Let's call the distance from the building to the base of the ladder "x".

According to the problem, the distance from the ground to the top of the ladder is 9 feet more than the distance from the building to the base of the ladder. So, the length of the ladder would be (x + 9).

Using the Pythagorean theorem, we can set up the following equation:

x^2 + (x + 9)^2 = 23^2.

Now, let's solve the equation step by step:

x^2 + (x + 9)^2 = 529. (23^2 = 529)

Expanding the equation:

x^2 + (x^2 + 18x + 81) = 529.

Combining like terms:

2x^2 + 18x + 81 = 529.

Subtracting 529 from both sides:

2x^2 + 18x - 448 = 0.

Now, we can solve this quadratic equation using factoring or the quadratic formula. In this case, factoring is a bit challenging, and the quadratic formula would be the easier choice.

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a,

where a = 2, b = 18, and c = -448.

Plugging the values into the quadratic formula:

x = (-18 ± √(18^2 - 4 * 2 * -448)) / (2 * 2).

Simplifying the expression under the square root:

x = (-18 ± √(324 + 3584)) / 4.

x = (-18 ± √(3908)) / 4.

Now, we need to determine the value of x. The problem asks for the distance up the building, so we can ignore the negative solution.

x = (-18 + √(3908)) / 4.

Using a calculator, find the value inside the square root:

x ≈ (-18 + √(3908)) / 4.

x ≈ (-18 + 62.5632) / 4.

x ≈ 44.5632 / 4.

x ≈ 11.1408.

Therefore, the ladder is approximately 11.1 feet up the building to the nearest tenth of a foot.