find x if x-3, 3x+5 and 18x-5 are three consecutive term of a geometric progression
(x - 3) r =(3 x + 5)
(3 x + 5) r = 18 x - 5
(3 x + 5)/(x - 3) = ( 18 x - 5) / (3 x + 5 )
(3 x + 5)^2 = (x - 3)( 18 x - 5)
9 x^2 + 30 x + 25 = 18 x^2 - 59 x + 15
9 x^2 - 89 x - 10 = 0
(x-10)(9x+1) = 0
x = 10 or x = -1/9
using x = 10
r = 35/7 = 5
or
r = 175/35 = 5
that works
use the properties of a GP
(3x+5)/(x-3) = (18x-5)/(3x+5)
(x-3)(18x-5) = (3x+5)^2
18x^2 - 69x + 15 = 9x^2 + 30x + 25
9x^2 - 89x - 10 = 0
(x-10)(9x + 1) = 0
x = 10 or x = -1/9
how can we find x in x-3, 3x+5 and 18x-5 geometric progression
Remember that there is a common ratio. That means that
(3x+5)/(x-3) = (18x-5)/(3x+5)
Now just solve for x.
It is okay
Well, it seems we have a geometric progression here. Let's find the common ratio (r) using the given terms.
For the first two terms:
(x - 3) / (3x + 5) = (3x + 5) / (18x - 5)
Now, let's get rid of the fractions by cross-multiplying:
(x - 3)(18x - 5) = (3x + 5)(3x + 5)
Expand and simplify both sides:
18x^2 - 53x + 15 = 9x^2 + 30x + 25
Combining like terms and moving everything to one side:
9x^2 + 83x - 10 = 0
Woah, that's a quadratic equation! Let's solve it:
Using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in the values:
x = (-83 ± sqrt(83^2 - 4 * 9 * -10)) / (2 * 9)
Calculating further:
x ≈ (-83 ± sqrt(6889 + 360)) / 18
x ≈ (-83 ± sqrt(7249)) / 18
Well, it seems the values of x are a little complicated. So, all I can say is, "Good luck finding the exact values of x!" Remember, clowns don't do math; we just crack jokes!
To find the value of x in this problem, we can use the definition of a geometric progression. In a geometric progression, each term is obtained by multiplying the previous term by a constant ratio.
So, let's set up the equation using the given information:
x - 3, 3x + 5, and 18x - 5 are three consecutive terms of a geometric progression.
The first term (a) is x - 3.
The second term (ar) is 3x + 5.
The third term (ar^2) is 18x - 5.
Using these terms, we can write two equations to solve for the value of x.
The first equation is (ar) = (x - 3)(r) = 3x + 5.
The second equation is (ar^2) = (3x + 5)(r) = 18x - 5.
Now, let's solve these equations:
(x - 3)(r) = 3x + 5
Expanding, we get: xr - 3r = 3x + 5
Simplifying and rearranging the terms, we get:
xr - 3r - 3x = 5
Similarly, for the second equation:
(3x + 5)(r) = 18x - 5
3xr + 5r = 18x - 5
Simplifying and rearranging the terms, we get:
3xr + 5r - 18x = -5
Now, we have a system of two equations:
xr - 3r - 3x = 5
3xr + 5r - 18x = -5
To solve this system, we can use substitution or elimination method. Let's use the substitution method:
From the first equation, we isolate r:
xr - 3r - 3x = 5
xr - 3r = 3x + 5
r(x - 3) = 3x + 5
r = (3x + 5)/(x - 3)
Now, substitute this value of r into the second equation:
3xr + 5r - 18x = -5
3x(3x + 5)/(x - 3) + 5(3x + 5)/(x - 3) - 18x = -5
To simplify this equation, let's find the common denominator:
(x - 3)
Multiplying each fraction by (x - 3), we get:
9x^2 + 15x + 15 + 15x + 25 - 18x(x - 3) = -5(x - 3)
Simplifying further, we get:
9x^2 + 15x + 15 + 15x + 25 - 18x^2 + 54x = -5x + 15
Combining like terms:
-9x^2 + 84x + 55 = -5x + 15
Rearranging the terms and setting the equation equal to zero:
-9x^2 + 89x + 40 = 0
Now, we can solve this quadratic equation. We can either factor it, complete the square, or use the quadratic formula to find the values of x.