8x^2+41x+42=0

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Solution for 8x^2+41x+42=0 equation:


Simplifying
8x2 + 41x + 42 = 0

Reorder the terms:
42 + 41x + 8x2 = 0

Solving
42 + 41x + 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'.
5.25 + 5.125x + x2 = 0

Move the constant term to the right:

Add '-5.25' to each side of the equation.
5.25 + 5.125x + -5.25 + x2 = 0 + -5.25

Reorder the terms:
5.25 + -5.25 + 5.125x + x2 = 0 + -5.25

Combine like terms: 5.25 + -5.25 = 0.00
0.00 + 5.125x + x2 = 0 + -5.25
5.125x + x2 = 0 + -5.25

Combine like terms: 0 + -5.25 = -5.25
5.125x + x2 = -5.25

The x term is 5.125x.  Take half its coefficient (2.5625).
Square it (6.56640625) and add it to both sides.

Add '6.56640625' to each side of the equation.
5.125x + 6.56640625 + x2 = -5.25 + 6.56640625

Reorder the terms:
6.56640625 + 5.125x + x2 = -5.25 + 6.56640625

Combine like terms: -5.25 + 6.56640625 = 1.31640625
6.56640625 + 5.125x + x2 = 1.31640625

Factor a perfect square on the left side:
(x + 2.5625)(x + 2.5625) = 1.31640625

Calculate the square root of the right side: 1.147347484

Break this problem into two subproblems by setting 
(x + 2.5625) equal to 1.147347484 and -1.147347484.

Subproblem 1

x + 2.5625 = 1.147347484 Simplifying x + 2.5625 = 1.147347484 Reorder the terms: 2.5625 + x = 1.147347484 Solving 2.5625 + x = 1.147347484 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5625' to each side of the equation. 2.5625 + -2.5625 + x = 1.147347484 + -2.5625 Combine like terms: 2.5625 + -2.5625 = 0.0000 0.0000 + x = 1.147347484 + -2.5625 x = 1.147347484 + -2.5625 Combine like terms: 1.147347484 + -2.5625 = -1.415152516 x = -1.415152516 Simplifying x = -1.415152516

Subproblem 2

x + 2.5625 = -1.147347484 Simplifying x + 2.5625 = -1.147347484 Reorder the terms: 2.5625 + x = -1.147347484 Solving 2.5625 + x = -1.147347484 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5625' to each side of the equation. 2.5625 + -2.5625 + x = -1.147347484 + -2.5625 Combine like terms: 2.5625 + -2.5625 = 0.0000 0.0000 + x = -1.147347484 + -2.5625 x = -1.147347484 + -2.5625 Combine like terms: -1.147347484 + -2.5625 = -3.709847484 x = -3.709847484 Simplifying x = -3.709847484

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.415152516, -3.709847484}

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