+4
Find all real numbers x such that (3x - 27)^3 + (27x - 3)^3 = (3x + 27x - 30)^3.
+2666 Simplify as follows:
\((3x - 27)^3 + (27x - 3)^3 = (3x + 27x - 30)^3\)
\((3x - 27)^3 + (27x - 3)^3 = (30x - 30)^3\)
\((3(x-9))^3 + (3(9x - 1))^3 = (3(10x - 10))^3\)
\((x-9)^3 + (9x-1)^3 = (10x-10)^3\)
\(-270x^3+2730x^2-2730x+270 = 0\)
\(9x^3 - 91x^2 +91x - 9 = 0\)
Notice that x = 1 is a solution to the equation. Now, divide the left-hand side by (x-1). The result will be quadratic, and you can use the quadratic formula to solve for the other 2 x values.
+618 We can factor the left-hand side of the equation as follows:
(3x - 27)^3 + (27x - 3)^3 = ((3x - 27) + (27x - 3))(((3x - 27)^2 - (3x - 27)(27x - 3) + (27x - 3)^2)
= (30x - 30)((3x - 27)^2 - (3x - 27)(27x - 3) + (27x - 3)^2).
We can factor the right-hand side of the equation as follows:
(3x + 27x - 30)^3 = (60x - 30)^3.
Since (60x−30)3=(30x−30)((60x−30)2), we can write the equation as follows:
(30x - 30)((3x - 27)^2 - (3x - 27)(27x - 3) + (27x - 3)^2) = (60x - 30)((60x - 30)^2).
Since (30x−30)=0, we can cancel it from both sides of the equation, which gives us:
((3x - 27)^2 - (3x - 27)(27x - 3) + (27x - 3)^2) = (60x - 30)^2.
Expanding both sides of the equation, we get:
9x^2 - 162x + 729 - 27x^3 + 196x^2 - 171x + 243 + 729x^2 - 189x + 9 = 3600x^2 - 7200x + 2700
Combining like terms, we get:
2547x^2 - 2316x - 2507 = 0
This quadratic factors as follows:
(x - 1)(2547x + 2507) = 0
The only real solution to this equation is x=1
+2666 Simplify as follows:
\((3x - 27)^3 + (27x - 3)^3 = (3x + 27x - 30)^3\)
\((3x - 27)^3 + (27x - 3)^3 = (30x - 30)^3\)
\((3(x-9))^3 + (3(9x - 1))^3 = (3(10x - 10))^3\)
\((x-9)^3 + (9x-1)^3 = (10x-10)^3\)
\(-270x^3+2730x^2-2730x+270 = 0\)
\(9x^3 - 91x^2 +91x - 9 = 0\)
Notice that x = 1 is a solution to the equation. Now, divide the left-hand side by (x-1). The result will be quadratic, and you can use the quadratic formula to solve for the other 2 x values.
