Question

A 13 foot ladder is leaning against a wall. If the height of the ladder against the wall is 2 feet more than twice the distance from the wall to the base of the ladder, find the height of the ladder.

Answer 1

height of ladder = 12 feet

let x be the distance from the wall to the base of the ladder then

height = 2x + 2

Using Pythagoras' identity

(2x + 2)² + x² = 13² ( expand bracket and simplify left side )

4x² + 8x + 4 + x² = 169 ( rearrange into standard form )

5x² + 8x - 165 = 0 ← in standard form

consider the factors of the product 5 × - 165 = - 825

which sum to the coefficient of the x- term + 8

These are + 33 and - 25

split the middle term using these factors

5x² - 25x + 33x - 165 = 0 ( factor by grouping )

5x(x - 5) + 33(x - 5) = 0 ( take out the common factor (x - 5) )

(x - 5)(5x + 33) = 0

equate each factor to zero and solve for x

x - 5 = 0 ⇒ x = 5

5x + 33 = 0 ⇒ x = - $\frac{33}{5}$

but x > 0 ⇒ x = 5

Hence height of ladder = (2 × 5 ) + 2 = 12 feet