9x^4-18x^3-22x^2-18x+9=0

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To solve this equation, we can first factor out a common factor of 9 from all the terms:

9(x^4 - 2x^3 - 2x^2 - 2x + 1) = 0

Now, we can look for factors of 1 (the constant term) that add up to -2 (the coefficient of the x^3 term). The factors that satisfy this condition are -1 and -1:

9(x^4 - x^3 - x^2 + x - x + 1) = 0

Now, we can factor by grouping:

9(x^3(x - 1) - x(x - 1) - 1(x - 1)) = 0

Factor out a common factor of (x - 1) from the terms inside the parentheses:

9((x^3 - x - 1)(x - 1)) = 0

The equation now becomes:

9(x^3 - x - 1)(x - 1) = 0

At this point, we have a product of three factors equal to zero. This means that one or more of the factors must be zero in order for the entire expression to equal zero.

Setting each factor equal to zero and solving for x gives us the solutions:

x^3 - x - 1 = 0
x - 1 = 0

Solving x - 1 = 0 for x gives us x = 1.

To solve x^3 - x - 1 = 0, we can use numerical methods such as Newton's method or use a computer algebra system to find approximate solutions.