8x^2+6x-19=0

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Solution for 8x^2+6x-19=0 equation:


Simplifying
8x2 + 6x + -19 = 0

Reorder the terms:
-19 + 6x + 8x2 = 0

Solving
-19 + 6x + 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 + 0.75x + x2 = 0

Move the constant term to the right:

Add '2.375' to each side of the equation.
-2.375 + 0.75x + 2.375 + x2 = 0 + 2.375

Reorder the terms:
-2.375 + 2.375 + 0.75x + x2 = 0 + 2.375

Combine like terms: -2.375 + 2.375 = 0.000
0.000 + 0.75x + x2 = 0 + 2.375
0.75x + x2 = 0 + 2.375

Combine like terms: 0 + 2.375 = 2.375
0.75x + x2 = 2.375

The x term is 0.75x.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75x + 0.140625 + x2 = 2.375 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = 2.375 + 0.140625

Combine like terms: 2.375 + 0.140625 = 2.515625
0.140625 + 0.75x + x2 = 2.515625

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 2.515625

Calculate the square root of the right side: 1.586072193

Break this problem into two subproblems by setting 
(x + 0.375) equal to 1.586072193 and -1.586072193.

Subproblem 1

x + 0.375 = 1.586072193 Simplifying x + 0.375 = 1.586072193 Reorder the terms: 0.375 + x = 1.586072193 Solving 0.375 + x = 1.586072193 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = 1.586072193 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = 1.586072193 + -0.375 x = 1.586072193 + -0.375 Combine like terms: 1.586072193 + -0.375 = 1.211072193 x = 1.211072193 Simplifying x = 1.211072193

Subproblem 2

x + 0.375 = -1.586072193 Simplifying x + 0.375 = -1.586072193 Reorder the terms: 0.375 + x = -1.586072193 Solving 0.375 + x = -1.586072193 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = -1.586072193 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = -1.586072193 + -0.375 x = -1.586072193 + -0.375 Combine like terms: -1.586072193 + -0.375 = -1.961072193 x = -1.961072193 Simplifying x = -1.961072193

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.211072193, -1.961072193}

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