4x^2-9=x

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


Simplifying
4x2 + -9 = x

Reorder the terms:
-9 + 4x2 = x

Solving
-9 + 4x2 = x

Solving for variable 'x'.

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

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

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

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

Move the constant term to the right:

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

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

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

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

The x term is -0.25x.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25x + 0.015625 + x2 = 2.25 + 0.015625

Reorder the terms:
0.015625 + -0.25x + x2 = 2.25 + 0.015625

Combine like terms: 2.25 + 0.015625 = 2.265625
0.015625 + -0.25x + x2 = 2.265625

Factor a perfect square on the left side:
(x + -0.125)(x + -0.125) = 2.265625

Calculate the square root of the right side: 1.505199322

Break this problem into two subproblems by setting 
(x + -0.125) equal to 1.505199322 and -1.505199322.

Subproblem 1

x + -0.125 = 1.505199322 Simplifying x + -0.125 = 1.505199322 Reorder the terms: -0.125 + x = 1.505199322 Solving -0.125 + x = 1.505199322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = 1.505199322 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = 1.505199322 + 0.125 x = 1.505199322 + 0.125 Combine like terms: 1.505199322 + 0.125 = 1.630199322 x = 1.630199322 Simplifying x = 1.630199322

Subproblem 2

x + -0.125 = -1.505199322 Simplifying x + -0.125 = -1.505199322 Reorder the terms: -0.125 + x = -1.505199322 Solving -0.125 + x = -1.505199322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = -1.505199322 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = -1.505199322 + 0.125 x = -1.505199322 + 0.125 Combine like terms: -1.505199322 + 0.125 = -1.380199322 x = -1.380199322 Simplifying x = -1.380199322

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.630199322, -1.380199322}

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