4x^2-3x-5=2x+6

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Solution for 4x^2-3x-5=2x+6 equation:


Simplifying
4x2 + -3x + -5 = 2x + 6

Reorder the terms:
-5 + -3x + 4x2 = 2x + 6

Reorder the terms:
-5 + -3x + 4x2 = 6 + 2x

Solving
-5 + -3x + 4x2 = 6 + 2x

Solving for variable 'x'.

Reorder the terms:
-5 + -6 + -3x + -2x + 4x2 = 6 + 2x + -6 + -2x

Combine like terms: -5 + -6 = -11
-11 + -3x + -2x + 4x2 = 6 + 2x + -6 + -2x

Combine like terms: -3x + -2x = -5x
-11 + -5x + 4x2 = 6 + 2x + -6 + -2x

Reorder the terms:
-11 + -5x + 4x2 = 6 + -6 + 2x + -2x

Combine like terms: 6 + -6 = 0
-11 + -5x + 4x2 = 0 + 2x + -2x
-11 + -5x + 4x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-11 + -5x + 4x2 = 0

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

Divide each side by '4'.
-2.75 + -1.25x + x2 = 0

Move the constant term to the right:

Add '2.75' to each side of the equation.
-2.75 + -1.25x + 2.75 + x2 = 0 + 2.75

Reorder the terms:
-2.75 + 2.75 + -1.25x + x2 = 0 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + -1.25x + x2 = 0 + 2.75
-1.25x + x2 = 0 + 2.75

Combine like terms: 0 + 2.75 = 2.75
-1.25x + x2 = 2.75

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 = 2.75 + 0.390625

Reorder the terms:
0.390625 + -1.25x + x2 = 2.75 + 0.390625

Combine like terms: 2.75 + 0.390625 = 3.140625
0.390625 + -1.25x + x2 = 3.140625

Factor a perfect square on the left side:
(x + -0.625)(x + -0.625) = 3.140625

Calculate the square root of the right side: 1.77218086

Break this problem into two subproblems by setting 
(x + -0.625) equal to 1.77218086 and -1.77218086.

Subproblem 1

x + -0.625 = 1.77218086 Simplifying x + -0.625 = 1.77218086 Reorder the terms: -0.625 + x = 1.77218086 Solving -0.625 + x = 1.77218086 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 = 1.77218086 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = 1.77218086 + 0.625 x = 1.77218086 + 0.625 Combine like terms: 1.77218086 + 0.625 = 2.39718086 x = 2.39718086 Simplifying x = 2.39718086

Subproblem 2

x + -0.625 = -1.77218086 Simplifying x + -0.625 = -1.77218086 Reorder the terms: -0.625 + x = -1.77218086 Solving -0.625 + x = -1.77218086 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 = -1.77218086 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = -1.77218086 + 0.625 x = -1.77218086 + 0.625 Combine like terms: -1.77218086 + 0.625 = -1.14718086 x = -1.14718086 Simplifying x = -1.14718086

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.39718086, -1.14718086}

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