Determine the value of x in the geometric sequence
9x + 8 ; 2x - 6 ; x - 8
To determine the value of x in the geometric sequence, we need to find the common ratio (r) and substitute it into the equation.
The common ratio (r) can be found by dividing any term in the sequence by the preceding term. Let's divide the second term (2x - 6) by the first term (9x + 8):
(2x - 6)/(9x + 8)
To determine if it is a geometric sequence, we need to see if this division results in a constant value.
Now, let's divide the third term (x - 8) by the second term (2x - 6):
(x - 8)/(2x - 6)
Again, we need to see if this division results in the same constant value as before.
Setting up the equation for the common ratio:
(2x - 6)/(9x + 8) = (x - 8)/(2x - 6)
To solve this equation, we can cross multiply:
(2x - 6)(2x - 6) = (9x + 8)(x - 8)
Expanding and simplifying:
4x^2 - 12x - 12x + 36 = 9x(x) + 9x(-8) + 8(x) + 8(-8)
4x^2 - 24x + 36 = 9x^2 - 72x + 8x - 64
Combining like terms:
4x^2 - 24x + 36 = 9x^2 - 64x - 64
Moving all terms to one side to set the equation equal to zero:
9x^2 - 64x - 64 - 4x^2 + 24x - 36 = 0
5x^2 - 40x - 100 = 0
Factoring:
5(x^2 - 8x - 20) = 0
Applying the zero product property:
x^2 - 8x - 20 = 0
To solve this quadratic equation, we can either factor or use the quadratic formula.
Factoring:
(x - 10)(x + 2) = 0
Setting each factor equal to zero:
x - 10 = 0 -> x = 10
x + 2 = 0 -> x = -2
Therefore, the possible values for x in the geometric sequence are x = 10 and x = -2.