A ladder 65 feet long is leaning against a wall. Its lower end is 25 feet away from the wall how much farther away will t be if the upper end is moved down 8 feet?

To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, the ladder acts as the hypotenuse, and the distance of the ladder's lower end from the wall is one side of the right triangle (25 feet). The other side is the distance that the ladder's upper end will be moved down (8 feet). Let's call the distance we need to find "x."

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

(25 + x)^2 = 65^2

First, let's simplify the equation by expanding:

625 + 50x + x^2 = 4225

Rearrange the equation to isolate the x^2 term:

x^2 + 50x + 625 - 4225 = 0

x^2 + 50x - 3600 = 0

Next, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In the equation x^2 + 50x - 3600 = 0:

a = 1, b = 50, c = -3600

Plugging in the values into the quadratic formula:

x = (-50 ± √(50^2 - 4(1)(-3600))) / (2(1))

x = (-50 ± √(2500 + 14400)) / 2

x = (-50 ± √16900) / 2

x = (-50 ± 130) / 2

Now, we can solve for both solutions:

x1 = (-50 + 130) / 2 = 40 feet

x2 = (-50 - 130) / 2 = -90 feet (disregard since distance cannot be negative in this context)

Therefore, if the upper end of the ladder is moved down by 8 feet, the ladder will be 40 feet farther away from the wall.