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Find all real values of x that satisfy (16x^2 - 9)^3 + (9x^2 - 16)^3 = (25x^2 - 25)^3.

 Dec 8, 2019
 #1
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Solve for x:
(9 x^2 - 16)^3 + (16 x^2 - 9)^3 = (25 x^2 - 25)^3

Expand out terms of the left hand side:
4825 x^6 - 10800 x^4 + 10800 x^2 - 4825 = (25 x^2 - 25)^3

Expand out terms of the right hand side:
4825 x^6 - 10800 x^4 + 10800 x^2 - 4825 = 15625 x^6 - 46875 x^4 + 46875 x^2 - 15625

Subtract 15625 x^6 - 46875 x^4 + 46875 x^2 - 15625 from both sides:
-10800 x^6 + 36075 x^4 - 36075 x^2 + 10800 = 0

The left hand side factors into a product with seven terms:
-75 (x - 1) (x + 1) (3 x - 4) (3 x + 4) (4 x - 3) (4 x + 3) = 0

Divide both sides by -75:
(x - 1) (x + 1) (3 x - 4) (3 x + 4) (4 x - 3) (4 x + 3) = 0

Split into six equations:
x - 1 = 0 or x + 1 = 0 or 3 x - 4 = 0 or 3 x + 4 = 0 or 4 x - 3 = 0 or 4 x + 3 = 0

Add 1 to both sides:
x = 1 or x + 1 = 0 or 3 x - 4 = 0 or 3 x + 4 = 0 or 4 x - 3 = 0 or 4 x + 3 = 0

Subtract 1 from both sides:
x = 1 or x = -1 or 3 x - 4 = 0 or 3 x + 4 = 0 or 4 x - 3 = 0 or 4 x + 3 = 0

Add 4 to both sides:
x = 1 or x = -1 or 3 x = 4 or 3 x + 4 = 0 or 4 x - 3 = 0 or 4 x + 3 = 0

Divide both sides by 3:
x = 1 or x = -1 or x = 4/3 or 3 x + 4 = 0 or 4 x - 3 = 0 or 4 x + 3 = 0

Subtract 4 from both sides:
x = 1 or x = -1 or x = 4/3 or 3 x = -4 or 4 x - 3 = 0 or 4 x + 3 = 0

Divide both sides by 3:
x = 1 or x = -1 or x = 4/3 or x = -4/3 or 4 x - 3 = 0 or 4 x + 3 = 0

Add 3 to both sides:
x = 1 or x = -1 or x = 4/3 or x = -4/3 or 4 x = 3 or 4 x + 3 = 0

Divide both sides by 4:
x = 1 or x = -1 or x = 4/3 or x = -4/3 or x = 3/4 or 4 x + 3 = 0

Subtract 3 from both sides:
x = 1 or x = -1 or x = 4/3 or x = -4/3 or x = 3/4 or 4 x = -3

Divide both sides by 4:
 
x = 1   or    x = -1    or    x = 4/3    or    x = -4/3    or    x = 3/4    or    x = -3/4

 Dec 9, 2019

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