4x^2=9x-1

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


Simplifying
4x2 = 9x + -1

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

Solving
4x2 = -1 + 9x

Solving for variable 'x'.

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

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

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

Combine like terms: 9x + -9x = 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 = 1.015625
1.265625 + -2.25x + x2 = 1.015625

Factor a perfect square on the left side:
(x + -1.125)(x + -1.125) = 1.015625

Calculate the square root of the right side: 1.007782219

Break this problem into two subproblems by setting 
(x + -1.125) equal to 1.007782219 and -1.007782219.

Subproblem 1

x + -1.125 = 1.007782219 Simplifying x + -1.125 = 1.007782219 Reorder the terms: -1.125 + x = 1.007782219 Solving -1.125 + x = 1.007782219 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 = 1.007782219 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = 1.007782219 + 1.125 x = 1.007782219 + 1.125 Combine like terms: 1.007782219 + 1.125 = 2.132782219 x = 2.132782219 Simplifying x = 2.132782219

Subproblem 2

x + -1.125 = -1.007782219 Simplifying x + -1.125 = -1.007782219 Reorder the terms: -1.125 + x = -1.007782219 Solving -1.125 + x = -1.007782219 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 = -1.007782219 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = -1.007782219 + 1.125 x = -1.007782219 + 1.125 Combine like terms: -1.007782219 + 1.125 = 0.117217781 x = 0.117217781 Simplifying x = 0.117217781

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.132782219, 0.117217781}

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