4x^2-9x=1

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Solution for 4x^2-9x=1 equation:


Simplifying
4x2 + -9x = 1

Reorder the terms:
-9x + 4x2 = 1

Solving
-9x + 4x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + -9x + 4x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -9x + 4x2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-0.25 + -2.25x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -2.25x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -2.25x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -2.25x + x2 = 0 + 0.25
-2.25x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-2.25x + x2 = 0.25

The x term is -2.25x.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25x + 1.265625 + x2 = 0.25 + 1.265625

Reorder the terms:
1.265625 + -2.25x + x2 = 0.25 + 1.265625

Combine like terms: 0.25 + 1.265625 = 1.515625
1.265625 + -2.25x + x2 = 1.515625

Factor a perfect square on the left side:
(x + -1.125)(x + -1.125) = 1.515625

Calculate the square root of the right side: 1.231107225

Break this problem into two subproblems by setting 
(x + -1.125) equal to 1.231107225 and -1.231107225.

Subproblem 1

x + -1.125 = 1.231107225 Simplifying x + -1.125 = 1.231107225 Reorder the terms: -1.125 + x = 1.231107225 Solving -1.125 + x = 1.231107225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + x = 1.231107225 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = 1.231107225 + 1.125 x = 1.231107225 + 1.125 Combine like terms: 1.231107225 + 1.125 = 2.356107225 x = 2.356107225 Simplifying x = 2.356107225

Subproblem 2

x + -1.125 = -1.231107225 Simplifying x + -1.125 = -1.231107225 Reorder the terms: -1.125 + x = -1.231107225 Solving -1.125 + x = -1.231107225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + x = -1.231107225 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = -1.231107225 + 1.125 x = -1.231107225 + 1.125 Combine like terms: -1.231107225 + 1.125 = -0.106107225 x = -0.106107225 Simplifying x = -0.106107225

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.356107225, -0.106107225}

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