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Find all real numbers x such that (3x - 27)^3 + (27x - 3)^3 = (3x + 27x - 30)^3.

 Oct 12, 2023

Best Answer 

 #4
avatar+2665 
+2

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. 

 Oct 12, 2023
 #1
avatar+423 
-1

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​

 Oct 12, 2023
 #2
avatar+4 
+2

Thanks for helping me

CoolMathUser123  Oct 12, 2023
 #3
avatar+33568 
+2

x = 1 is certainly a solution, but there are two other real number solutions to the original cubic equation!

Alan  Oct 12, 2023
 #4
avatar+2665 
+2
Best Answer

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. 

BuilderBoi Oct 12, 2023

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