19x^2+5x-4=0

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Solution for 19x^2+5x-4=0 equation:


Simplifying
19x2 + 5x + -4 = 0

Reorder the terms:
-4 + 5x + 19x2 = 0

Solving
-4 + 5x + 19x2 = 0

Solving for variable 'x'.

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

Divide each side by '19'.
-0.2105263158 + 0.2631578947x + x2 = 0

Move the constant term to the right:

Add '0.2105263158' to each side of the equation.
-0.2105263158 + 0.2631578947x + 0.2105263158 + x2 = 0 + 0.2105263158

Reorder the terms:
-0.2105263158 + 0.2105263158 + 0.2631578947x + x2 = 0 + 0.2105263158

Combine like terms: -0.2105263158 + 0.2105263158 = 0.0000000000
0.0000000000 + 0.2631578947x + x2 = 0 + 0.2105263158
0.2631578947x + x2 = 0 + 0.2105263158

Combine like terms: 0 + 0.2105263158 = 0.2105263158
0.2631578947x + x2 = 0.2105263158

The x term is 0.2631578947x.  Take half its coefficient (0.1315789474).
Square it (0.01731301940) and add it to both sides.

Add '0.01731301940' to each side of the equation.
0.2631578947x + 0.01731301940 + x2 = 0.2105263158 + 0.01731301940

Reorder the terms:
0.01731301940 + 0.2631578947x + x2 = 0.2105263158 + 0.01731301940

Combine like terms: 0.2105263158 + 0.01731301940 = 0.2278393352
0.01731301940 + 0.2631578947x + x2 = 0.2278393352

Factor a perfect square on the left side:
(x + 0.1315789474)(x + 0.1315789474) = 0.2278393352

Calculate the square root of the right side: 0.477325188

Break this problem into two subproblems by setting 
(x + 0.1315789474) equal to 0.477325188 and -0.477325188.

Subproblem 1

x + 0.1315789474 = 0.477325188 Simplifying x + 0.1315789474 = 0.477325188 Reorder the terms: 0.1315789474 + x = 0.477325188 Solving 0.1315789474 + x = 0.477325188 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1315789474' to each side of the equation. 0.1315789474 + -0.1315789474 + x = 0.477325188 + -0.1315789474 Combine like terms: 0.1315789474 + -0.1315789474 = 0.0000000000 0.0000000000 + x = 0.477325188 + -0.1315789474 x = 0.477325188 + -0.1315789474 Combine like terms: 0.477325188 + -0.1315789474 = 0.3457462406 x = 0.3457462406 Simplifying x = 0.3457462406

Subproblem 2

x + 0.1315789474 = -0.477325188 Simplifying x + 0.1315789474 = -0.477325188 Reorder the terms: 0.1315789474 + x = -0.477325188 Solving 0.1315789474 + x = -0.477325188 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1315789474' to each side of the equation. 0.1315789474 + -0.1315789474 + x = -0.477325188 + -0.1315789474 Combine like terms: 0.1315789474 + -0.1315789474 = 0.0000000000 0.0000000000 + x = -0.477325188 + -0.1315789474 x = -0.477325188 + -0.1315789474 Combine like terms: -0.477325188 + -0.1315789474 = -0.6089041354 x = -0.6089041354 Simplifying x = -0.6089041354

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.3457462406, -0.6089041354}

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