Question

In parallelogram ABCD , diagonals AC¯¯¯¯¯ and BD¯¯¯¯¯ intersect at point E, BE=2x2−x , and DE=x2+6 . What is BD ? Enter your answer in the box. units

Answer 1

First thing you need to know is that the diagonals of a parallelogram bisect each other.
Since AC and BD (the two diagonals) intersect at point E, this means that E is the midpoint of BD, therefore:
BE = DE

We are given that:
BE = 2x^2 - x
DE = x^2 + 6

Since BE = DE, therefore:
2x^2 - x = x^2 + 6
2x^2 - x^2 - x - 6 = 0
x^2 - x - 6 = 0
(x-3)(x+2) = 0
Either x = 3
or x = -2
Since no constraints are given for the value of x, therefore both values can be accepted and used to calculate BD as shown below.

Choice #1:
At x = 3
BE = 2(3)^2 - 3 = 15 units
DE = (3)^2 + 6 = 15 units
BD = 15+15 = 30 units

Choice #2:
At x = -2:
BE = 2(-2)^2 -- 2 = 2(4) + 2 = 10 units
DE = (-2)^2 + 6 = 10 units
BD = 10 + 10 = 20 units

Based on the above calculations:
BD can either be 20 units or 30 units