Final answer:
The possible values of x in the parallelogram SHIP, where diagonals intersect at B and given HR = 3x+40 and RP = x^2, are found by setting up an equation x^2 = 2(3x + 40) and then factoring to find x = 10 and x = -8.
Step-by-step explanation:
The question involves finding the value of x in the context of a geometry problem involving a parallelogram named SHIP, where the diagonals intersect at point B. Given are two expressions that represent the lengths of segments HR and RP of the parallelogram's diagonals: HR = 3x + 40 and RP = x2. In a parallelogram, the diagonals bisect each other, so the lengths of HR and RB are equal; therefore, RP is also equal to 2HR. Setting the equations equal to each other because they represent the same length, we can find the value of x.
The equation becomes x2 = 2(3x + 40), which simplifies to x2 - 6x - 80 = 0. To solve the quadratic equation, we can factor if possible, use the quadratic formula, or complete the square. Upon factoring, we find that (x - 10)(x + 8) = 0. Therefore, the possible values for x are 10 and -8.