4x^2-5x-2=1

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


Simplifying
4x2 + -5x + -2 = 1

Reorder the terms:
-2 + -5x + 4x2 = 1

Solving
-2 + -5x + 4x2 = 1

Solving for variable 'x'.

Reorder the terms:
-2 + -1 + -5x + 4x2 = 1 + -1

Combine like terms: -2 + -1 = -3
-3 + -5x + 4x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-3 + -5x + 4x2 = 0

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

Divide each side by '4'.
-0.75 + -1.25x + x2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + -1.25x + 0.75 + x2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + -1.25x + x2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + -1.25x + x2 = 0 + 0.75
-1.25x + x2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
-1.25x + x2 = 0.75

The x term is -1.25x.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25x + 0.390625 + x2 = 0.75 + 0.390625

Reorder the terms:
0.390625 + -1.25x + x2 = 0.75 + 0.390625

Combine like terms: 0.75 + 0.390625 = 1.140625
0.390625 + -1.25x + x2 = 1.140625

Factor a perfect square on the left side:
(x + -0.625)(x + -0.625) = 1.140625

Calculate the square root of the right side: 1.068000468

Break this problem into two subproblems by setting 
(x + -0.625) equal to 1.068000468 and -1.068000468.

Subproblem 1

x + -0.625 = 1.068000468 Simplifying x + -0.625 = 1.068000468 Reorder the terms: -0.625 + x = 1.068000468 Solving -0.625 + x = 1.068000468 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = 1.068000468 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = 1.068000468 + 0.625 x = 1.068000468 + 0.625 Combine like terms: 1.068000468 + 0.625 = 1.693000468 x = 1.693000468 Simplifying x = 1.693000468

Subproblem 2

x + -0.625 = -1.068000468 Simplifying x + -0.625 = -1.068000468 Reorder the terms: -0.625 + x = -1.068000468 Solving -0.625 + x = -1.068000468 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = -1.068000468 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = -1.068000468 + 0.625 x = -1.068000468 + 0.625 Combine like terms: -1.068000468 + 0.625 = -0.443000468 x = -0.443000468 Simplifying x = -0.443000468

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.693000468, -0.443000468}

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