(4x^2-4x+23)+(4x^2+8x-42)=

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Solution for (4x^2-4x+23)+(4x^2+8x-42)= equation:


Simplifying
(4x2 + -4x + 23) + (4x2 + 8x + -42) = 0

Reorder the terms:
(23 + -4x + 4x2) + (4x2 + 8x + -42) = 0

Remove parenthesis around (23 + -4x + 4x2)
23 + -4x + 4x2 + (4x2 + 8x + -42) = 0

Reorder the terms:
23 + -4x + 4x2 + (-42 + 8x + 4x2) = 0

Remove parenthesis around (-42 + 8x + 4x2)
23 + -4x + 4x2 + -42 + 8x + 4x2 = 0

Reorder the terms:
23 + -42 + -4x + 8x + 4x2 + 4x2 = 0

Combine like terms: 23 + -42 = -19
-19 + -4x + 8x + 4x2 + 4x2 = 0

Combine like terms: -4x + 8x = 4x
-19 + 4x + 4x2 + 4x2 = 0

Combine like terms: 4x2 + 4x2 = 8x2
-19 + 4x + 8x2 = 0

Solving
-19 + 4x + 8x2 = 0

Solving for variable 'x'.

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

Divide each side by '8'.
-2.375 + 0.5x + x2 = 0

Move the constant term to the right:

Add '2.375' to each side of the equation.
-2.375 + 0.5x + 2.375 + x2 = 0 + 2.375

Reorder the terms:
-2.375 + 2.375 + 0.5x + x2 = 0 + 2.375

Combine like terms: -2.375 + 2.375 = 0.000
0.000 + 0.5x + x2 = 0 + 2.375
0.5x + x2 = 0 + 2.375

Combine like terms: 0 + 2.375 = 2.375
0.5x + x2 = 2.375

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 2.375 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 2.375 + 0.0625

Combine like terms: 2.375 + 0.0625 = 2.4375
0.0625 + 0.5x + x2 = 2.4375

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 2.4375

Calculate the square root of the right side: 1.5612495

Break this problem into two subproblems by setting 
(x + 0.25) equal to 1.5612495 and -1.5612495.

Subproblem 1

x + 0.25 = 1.5612495 Simplifying x + 0.25 = 1.5612495 Reorder the terms: 0.25 + x = 1.5612495 Solving 0.25 + x = 1.5612495 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 1.5612495 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 1.5612495 + -0.25 x = 1.5612495 + -0.25 Combine like terms: 1.5612495 + -0.25 = 1.3112495 x = 1.3112495 Simplifying x = 1.3112495

Subproblem 2

x + 0.25 = -1.5612495 Simplifying x + 0.25 = -1.5612495 Reorder the terms: 0.25 + x = -1.5612495 Solving 0.25 + x = -1.5612495 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -1.5612495 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -1.5612495 + -0.25 x = -1.5612495 + -0.25 Combine like terms: -1.5612495 + -0.25 = -1.8112495 x = -1.8112495 Simplifying x = -1.8112495

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3112495, -1.8112495}

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