Solve the equation without using a calculator (x^2-9x-1)^(10)+99x^(10)=10x^9(x^2-1)

Question

Solve the equation without using a calculator


`(x^2-9x-1)^(10)+99x^(10)=10x^9(x^2-1)`

Answer 1

• x = 5 ± √26

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Convert the given equation as below:

• 
  `(x^2-9x-1)^(10)+99x^(10)=10x^9(x^2-1)`
• 
  `(x^2-9x-1)^(10)+99x^(10)=10x^9(x^2-9x-1)+10x^9(9x)`
• 
  `(x^2-9x-1)^(10)+99x^(10)=10x^9(x^2-9x-1)+90x^(10)`
• 
  `(x^2-9x-1)^(10)+9x^(10)=10x^9(x^2-9x-1)`
• 
  `(x^2-9x-1)^(10)-9x^9(x^2-9x-1)=x^9(x^2-9x-1)-9x^9x`

Let's substitute x² - 9x - 1 = t, then the equation is:

• 
  `t^9t-9x^9t=x^9t-9x^9x`

Comparing both sides, we see that x = t, so:

• x² - 9x - 1 = x
• x² - 10x - 1 = 0

Solving this quadratic equation we get:

• x = 5 ± √26