(2x-5)(x+4)-1=0

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Solution for (2x-5)(x+4)-1=0 equation: 


Simplifying
(2x + -5)(x + 4) + -1 = 0

Reorder the terms:
(-5 + 2x)(x + 4) + -1 = 0

Reorder the terms:
(-5 + 2x)(4 + x) + -1 = 0

Multiply (-5 + 2x) * (4 + x)
(-5(4 + x) + 2x * (4 + x)) + -1 = 0
((4 * -5 + x * -5) + 2x * (4 + x)) + -1 = 0
((-20 + -5x) + 2x * (4 + x)) + -1 = 0
(-20 + -5x + (4 * 2x + x * 2x)) + -1 = 0
(-20 + -5x + (8x + 2x2)) + -1 = 0

Combine like terms: -5x + 8x = 3x
(-20 + 3x + 2x2) + -1 = 0

Reorder the terms:
-20 + -1 + 3x + 2x2 = 0

Combine like terms: -20 + -1 = -21
-21 + 3x + 2x2 = 0

Solving
-21 + 3x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-10.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + 1.5x + 10.5 + x2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + 1.5x + x2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + 1.5x + x2 = 0 + 10.5
1.5x + x2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
1.5x + x2 = 10.5

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 10.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 10.5 + 0.5625

Combine like terms: 10.5 + 0.5625 = 11.0625
0.5625 + 1.5x + x2 = 11.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 11.0625

Calculate the square root of the right side: 3.326033674

Break this problem into two subproblems by setting 
(x + 0.75) equal to 3.326033674 and -3.326033674.


Subproblem 1
x + 0.75 = 3.326033674

Simplifying
x + 0.75 = 3.326033674

Reorder the terms:
0.75 + x = 3.326033674

Solving
0.75 + x = 3.326033674

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 3.326033674 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 3.326033674 + -0.75
x = 3.326033674 + -0.75

Combine like terms: 3.326033674 + -0.75 = 2.576033674
x = 2.576033674

Simplifying
x = 2.576033674


Subproblem 2
x + 0.75 = -3.326033674

Simplifying
x + 0.75 = -3.326033674

Reorder the terms:
0.75 + x = -3.326033674

Solving
0.75 + x = -3.326033674

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -3.326033674 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -3.326033674 + -0.75
x = -3.326033674 + -0.75

Combine like terms: -3.326033674 + -0.75 = -4.076033674
x = -4.076033674

Simplifying
x = -4.076033674


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.576033674, -4.076033674}

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