x-1=4x^2+5x-6

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Solution for x-1=4x^2+5x-6 equation:


Simplifying
x + -1 = 4x2 + 5x + -6

Reorder the terms:
-1 + x = 4x2 + 5x + -6

Reorder the terms:
-1 + x = -6 + 5x + 4x2

Solving
-1 + x = -6 + 5x + 4x2

Solving for variable 'x'.

Reorder the terms:
-1 + 6 + x + -5x + -4x2 = -6 + 5x + 4x2 + 6 + -5x + -4x2

Combine like terms: -1 + 6 = 5
5 + x + -5x + -4x2 = -6 + 5x + 4x2 + 6 + -5x + -4x2

Combine like terms: x + -5x = -4x
5 + -4x + -4x2 = -6 + 5x + 4x2 + 6 + -5x + -4x2

Reorder the terms:
5 + -4x + -4x2 = -6 + 6 + 5x + -5x + 4x2 + -4x2

Combine like terms: -6 + 6 = 0
5 + -4x + -4x2 = 0 + 5x + -5x + 4x2 + -4x2
5 + -4x + -4x2 = 5x + -5x + 4x2 + -4x2

Combine like terms: 5x + -5x = 0
5 + -4x + -4x2 = 0 + 4x2 + -4x2
5 + -4x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
5 + -4x + -4x2 = 0

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

Divide each side by '-4'.
-1.25 + x + x2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + x + 1.25 + x2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + x + x2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x + x2 = 0 + 1.25
x + x2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
x + x2 = 1.25

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 1.25 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 1.25 + 0.25

Combine like terms: 1.25 + 0.25 = 1.5
1.25 + x2 = 1.5

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.224744871 and -1.224744871.

Subproblem 1

x + 0.5 = 1.224744871 Simplifying x + 0.5 = 1.224744871 Reorder the terms: 0.5 + x = 1.224744871 Solving 0.5 + x = 1.224744871 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 1.224744871 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.224744871 + -0.5 x = 1.224744871 + -0.5 Combine like terms: 1.224744871 + -0.5 = 0.724744871 x = 0.724744871 Simplifying x = 0.724744871

Subproblem 2

x + 0.5 = -1.224744871 Simplifying x + 0.5 = -1.224744871 Reorder the terms: 0.5 + x = -1.224744871 Solving 0.5 + x = -1.224744871 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -1.224744871 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.224744871 + -0.5 x = -1.224744871 + -0.5 Combine like terms: -1.224744871 + -0.5 = -1.724744871 x = -1.724744871 Simplifying x = -1.724744871

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.724744871, -1.724744871}

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