(4x-1)(x+3)-5=0

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Solution for (4x-1)(x+3)-5=0 equation:


Simplifying
(4x + -1)(x + 3) + -5 = 0

Reorder the terms:
(-1 + 4x)(x + 3) + -5 = 0

Reorder the terms:
(-1 + 4x)(3 + x) + -5 = 0

Multiply (-1 + 4x) * (3 + x)
(-1(3 + x) + 4x * (3 + x)) + -5 = 0
((3 * -1 + x * -1) + 4x * (3 + x)) + -5 = 0
((-3 + -1x) + 4x * (3 + x)) + -5 = 0
(-3 + -1x + (3 * 4x + x * 4x)) + -5 = 0
(-3 + -1x + (12x + 4x2)) + -5 = 0

Combine like terms: -1x + 12x = 11x
(-3 + 11x + 4x2) + -5 = 0

Reorder the terms:
-3 + -5 + 11x + 4x2 = 0

Combine like terms: -3 + -5 = -8
-8 + 11x + 4x2 = 0

Solving
-8 + 11x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-2 + 2.75x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 2.75x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + 2.75x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 2.75x + x2 = 0 + 2
2.75x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
2.75x + x2 = 2

The x term is 2.75x.  Take half its coefficient (1.375).
Square it (1.890625) and add it to both sides.

Add '1.890625' to each side of the equation.
2.75x + 1.890625 + x2 = 2 + 1.890625

Reorder the terms:
1.890625 + 2.75x + x2 = 2 + 1.890625

Combine like terms: 2 + 1.890625 = 3.890625
1.890625 + 2.75x + x2 = 3.890625

Factor a perfect square on the left side:
(x + 1.375)(x + 1.375) = 3.890625

Calculate the square root of the right side: 1.97246673

Break this problem into two subproblems by setting 
(x + 1.375) equal to 1.97246673 and -1.97246673.

Subproblem 1

x + 1.375 = 1.97246673 Simplifying x + 1.375 = 1.97246673 Reorder the terms: 1.375 + x = 1.97246673 Solving 1.375 + x = 1.97246673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = 1.97246673 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = 1.97246673 + -1.375 x = 1.97246673 + -1.375 Combine like terms: 1.97246673 + -1.375 = 0.59746673 x = 0.59746673 Simplifying x = 0.59746673

Subproblem 2

x + 1.375 = -1.97246673 Simplifying x + 1.375 = -1.97246673 Reorder the terms: 1.375 + x = -1.97246673 Solving 1.375 + x = -1.97246673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = -1.97246673 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = -1.97246673 + -1.375 x = -1.97246673 + -1.375 Combine like terms: -1.97246673 + -1.375 = -3.34746673 x = -3.34746673 Simplifying x = -3.34746673

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.59746673, -3.34746673}

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