A 13 foot ladder is leaning against a vertical wall. The distance from the top of the ladder to the bottom of the wall is 7 feet longer than the distance from the bottom of the ladder to the bottom of the wall. Find how high the top of the ladder reacher on the side of the wall.

a= x

b = x+7
c = 13

x^2 + (x+7)^2 = 13^2
= x^2 + x^2 +7x +7x + 49 = 169
= 2x^2 + 14x + 49 = 169
= 2x^2 + 14x + 49 -169= 169-169
= 2x^2 + 14x -120 = 0
= x^2 +7x -60 = 0
(x -5)(x+12)= 0

x = 5
x = -12 ignored when it is negative

a= 5
b = 12

Well, it sounds like this ladder sure knows how to reach for the stars! Let's try to figure out how high it's reaching on the side of the wall.

Let's call the distance from the bottom of the ladder to the bottom of the wall "x." According to the problem, the distance from the top of the ladder to the bottom of the wall is 7 feet longer than x. So, we can say that the height on the side of the wall is x + 7.

Now, we know that the ladder is 13 feet long. This forms a right triangle with the ladder as the hypotenuse, and the x and (x + 7) sides as the legs.

Using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can set up an equation:

x^2 + (x+7)^2 = 13^2

Simplifying that, we get:

x^2 + x^2 + 14x + 49 = 169

Combining like terms, we have:

2x^2 + 14x - 120 = 0

Solving that quadratic equation, we find that x = 5.

So, the distance from the bottom of the ladder to the bottom of the wall is 5 feet, and the height on the side of the wall is x + 7, which is 5 + 7 = 12 feet.

Voila! The top of the ladder reaches 12 feet on the side of the wall.

Let's assume the distance from the bottom of the ladder to the bottom of the wall is x feet.

According to the given information, the distance from the top of the ladder to the bottom of the wall is 7 feet longer than the distance from the bottom of the ladder to the bottom of the wall. So, this distance can be represented as x + 7 feet.

We can use the Pythagorean theorem to find the height the top of the ladder reaches on the side of the wall. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (in this case the ladder) is equal to the sum of the squares of the other two sides.

In our case, the ladder is the hypotenuse, the distance from the bottom of the ladder to the bottom of the wall is one side (x), and the distance from the top of the ladder to the bottom of the wall is the other side (x + 7).

So, according to the Pythagorean theorem:
x^2 + (x + 7)^2 = 13^2

Simplifying:
x^2 + (x^2 + 14x + 49) = 169
2x^2 + 14x + 49 = 169
2x^2 + 14x - 120 = 0

Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Since it can be easily factored, let's factor it:
2x^2 + 14x - 120 = 0
2(x^2 + 7x - 60) = 0
2(x + 12)(x - 5) = 0

Setting each factor equal to zero, we have two possible values for x:
x + 12 = 0 or x - 5 = 0

Solving for x:
For x + 12 = 0:
x = -12

For x - 5 = 0:
x = 5

Since distance cannot be negative, we ignore x = -12.

Therefore, the distance from the bottom of the ladder to the bottom of the wall is x = 5 feet.

To find the height the top of the ladder reaches on the side of the wall (h), we substitute x = 5 into h = x + 7:
h = 5 + 7 = 12 feet.

So, the top of the ladder reaches a height of 12 feet on the side of the wall.

To solve this problem, let's first define the variables:

Let x be the distance from the bottom of the ladder to the bottom of the wall.
Then, x + 7 represents the distance from the top of the ladder to the bottom of the wall.

We can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the ladder acts as the hypotenuse. So, we have:

x^2 + (x + 7)^2 = 13^2

Simplifying the equation, we obtain:

x^2 + (x^2 + 14x + 49) = 169
2x^2 + 14x + 49 = 169
2x^2 + 14x - 120 = 0

Now we have a quadratic equation, which we can solve by factoring, completing the square, or using the quadratic formula. Let's use factoring to solve it:

2(x^2 + 7x - 60) = 0
2(x + 12)(x - 5) = 0

Setting each factor equal to zero:

x + 12 = 0 or x - 5 = 0

Solving for x gives us two possible solutions:

x = -12 or x = 5

Since we are dealing with a positive distance, we discard the negative value. Thus, x = 5.

Therefore, the distance from the bottom of the ladder to the bottom of the wall is 5 feet, and the distance from the top of the ladder to the bottom of the wall is 5 + 7 = 12 feet.

Hence, the top of the ladder reaches 12 feet on the side of the wall.