5x^2+8x-165
A 13-foot ladder is placed so that it reaches to a point on the
wall that is 2 feet higher than twice the distance from the base of
the wall to the base of the ladder.
We have x feet from the wall and x + 2 feet up the wall.
x^2 + (2x + 2)^2 = 13^2
x^2 + (4x^2 + 4x + 4x + 4) = 13^2
5x^2 + 8x + 4 = 169
5x^2 + 8x + 4 - 169 = 0
5x^2 + 8x - 165 = 0
From here we use the quadratic formula to find any roots.
-b ± √(b² - 4*a*c)
x = ----------------------
2a
-8 ± √(8² - 4*5*(-165))
x = ----------------------------
2(5)
-8 ± √(64 - (-3300))
x = -------------------------
10
-8 ± √(3364)
x = -----------------
10
-8 ± 58
x = ----------
10
-8 + 58 -8 - 58
x = ---------- and x = ----------
10 10
50 -66
x = ---- and x = -----
10 10
x = 5 and x = -6.6
We can ignore the negative value. It doesn't make
sense that the ladder was placed -6.6 feet from the wall.
The answer is that the ladder was placed 5 feet from the wall
and reached 12 feet up the wall.