I was solving for x using factoring since you did not specify. If you need complete the square, let me know.
Solve for x:
((x^2 - 2 x - 35) (4 x^3 - 9 x))/((7 x - 49) (2 x^3 - 3 x^2)) = 0
Multiply both sides by (7 x - 49) (2 x^3 - 3 x^2):
(x^2 - 2 x - 35) (4 x^3 - 9 x) = 0
Split into two equations:
x^2 - 2 x - 35 = 0 or 4 x^3 - 9 x = 0
The left hand side factors into a product with two terms:
(x - 7) (x + 5) = 0 or 4 x^3 - 9 x = 0
Split into two equations:
x - 7 = 0 or x + 5 = 0 or 4 x^3 - 9 x = 0
Add 7 to both sides:
x = 7 or x + 5 = 0 or 4 x^3 - 9 x = 0
Subtract 5 from both sides:
x = 7 or x = -5 or 4 x^3 - 9 x = 0
The left hand side factors into a product with three terms:
x = 7 or x = -5 or x (2 x - 3) (2 x + 3) = 0
Split into three equations:
x = 7 or x = -5 or x = 0 or 2 x - 3 = 0 or 2 x + 3 = 0
Add 3 to both sides:
x = 7 or x = -5 or x = 0 or 2 x = 3 or 2 x + 3 = 0
Divide both sides by 2:
x = 7 or x = -5 or x = 0 or x = 3/2 or 2 x + 3 = 0
Subtract 3 from both sides:
x = 7 or x = -5 or x = 0 or x = 3/2 or 2 x = -3
Divide both sides by 2:
x = 7 or x = -5 or x = 0 or x = 3/2 or x = -3/2
((x^2 - 2 x - 35) (4 x^3 - 9 x))/((7 x - 49) (2 x^3 - 3 x^2)) ⇒ ((-35 - 2 (-5) + (-5)^2) (-9 (-5) + 4 (-5)^3))/((-49 + 7 (-5)) (-3 (-5)^2 + 2 (-5)^3)) = 0:
So this solution is correct
((x^2 - 2 x - 35) (4 x^3 - 9 x))/((7 x - 49) (2 x^3 - 3 x^2)) ⇒ ((-35 - (2 (-3))/2 + (-3/2)^2) (-(9 (-3))/2 + 4 (-3/2)^3))/((-49 + (7 (-3))/2) (-3 (-3/2)^2 + 2 (-3/2)^3)) = 0:
So this solution is correct
((x^2 - 2 x - 35) (4 x^3 - 9 x))/((7 x - 49) (2 x^3 - 3 x^2)) ⇒ ((-35 - 2 0 + 0^2) (-9 0 + 4 0^3))/((7 0 - 49) (-3 0^2 + 2 0^3)) = (undefined):
So this solution is incorrect
((x^2 - 2 x - 35) (4 x^3 - 9 x))/((7 x - 49) (2 x^3 - 3 x^2)) ⇒ ((-35 - (2 3)/2 + (3/2)^2) (4 (3/2)^3 - (9 3)/2))/(((7 3)/2 - 49) (2 (3/2)^3 - 3 (3/2)^2)) = (undefined):
So this solution is incorrect
((x^2 - 2 x - 35) (4 x^3 - 9 x))/((7 x - 49) (2 x^3 - 3 x^2)) ⇒ ((-35 - 2 7 + 7^2) (4 7^3 - 9 7))/((7 7 - 49) (2 7^3 - 3 7^2)) = (undefined):
So this solution is incorrect
The solutions are:
Answer: x = -5 or x = -3/2
