SOLUTION: A ladder is leaning against a wall. The distance from the bottom of the ladder to the wall is 3 feet less than the length of the ladder. Find the length of the ladder, if the dista

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Question 165009: A ladder is leaning against a wall. The distance from the bottom of the ladder to the wall is 3 feet less than the length of the ladder. Find the length of the ladder, if the distance of its top to the ground is 4 feet less than the length of the ladder.

Answer by nerdybill(7384) (Show Source):
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Let x = length of the ladder
x-3 = distance from the bottom of the ladder to the wall
x-4 = distance of its top to the ground
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Since the wall the the ground forms a right angle, this is a "right triangle". We can apply Pythagorean theorem:
(x-3)^2 + (x-4)^2 = x^2
x^2-6x+9 + x^2-8x+16 = x^2
2x^2-14x+25 = x^2
x^2-14x+25 = 0
Since we can't factor, we must resort to the quadratic formula. Doing so, will result in:
x = {11.90, 2.10}
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We can toss out the 2.10 answer leaving us with:
11.90 feet (length of the ladder)
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Details of quadratic:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=96 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 11.8989794855664, 2.10102051443364. Here's your graph: