A ladder is two feet longer than the height of a certain wall. When the top of the ladder is placed against the top of the wall, the distance from the base of the ladder to the bottom of the wall is equal to the height of the wall. How high is the wall?
Let x = height of wall and use Pythagorean theorem.
x^2 + x^2 = (x+2)^2 = x^2 + 4x + 4
Move all terms to left hand side of equation.
x^2 - 4x - 4 = 0
Unfortunately, I don't see how to factor this.
To find the height of the wall, let's denote it as 'h'. According to the problem, the ladder is two feet longer than the height of the wall. Therefore, the length of the ladder can be expressed as 'h + 2'.
When the ladder is placed against the wall, forming a right-angled triangle, we have the base of the ladder as the distance from the bottom of the wall to the base of the ladder. This is equal to the height of the wall.
Using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides (the height of the wall and the base of the ladder), we can set up the following equation:
(base of the ladder)^2 + (height of the wall)^2 = (length of the ladder)^2
Since the base of the ladder is equal to the height of the wall, we can rewrite the equation as:
(h)^2 + (h)^2 = (h + 2)^2
Now, we can simplify the equation and solve for 'h':
h^2 + h^2 = (h + 2)^2
2h^2 = h^2 + 4h + 4
h^2 - 4h - 4 = 0
To solve this quadratic equation, we can use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -4, and c = -4. Plugging these values into the quadratic formula:
h = (-(-4) ± √((-4)^2 - 4*1*-4)) / (2*1)
h = (4 ± √(16 + 16)) / 2
h = (4 ± √32) / 2
h = (4 ± 4√2) / 2
h = 2 ± 2√2
Since a measurement of height cannot be negative, we discard the negative value and take the positive value:
h = 2 + 2√2
Therefore, the height of the wall is approximately 4.83 feet.