Recall that Area of Triangle = (1/2) *base* height.
Let the height = x.
And statement said base = 8 ft longer than height = (x+8)
Therefore: A = (1/2)x(x+8)
384 = x(x+8)/2 Cross Multiplying
768 = x^2 + 8x Re arrange, this is a quadratic equation.
x^2 + 8x = 768
x^2 + 8x - 768 = 0. For quadratic factoring we think of two numbers that would multiply to give -768 and add up to give +8.
By search, the numbers are 32 and -24. 32*-24 = -768 and (32+ -24) = 32-24 =8.
Solving:
x^2 + 8x - 768 = 0 Replace 8x with (32x -24x)
x^2 + 32x -24x -768 =0
x(x+32) -24(x+32) = 0
(x+32)(x -24) = 0 Therefore (x+32) = 0 x = 0-32 =-32.
(x-24) = 0 x = 0+24 =24.
x = -32 or 24.
But our length can not be negative. Therefore x = 24. We throw away the negative answer.
Recall height = x = 24 ft.
Base = (x +8) = (24+8) = 32 ft.
Base = 32 ft, Height = 8ft. That's our answer. Hurray!
