(19x+3)*x=14

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Solution for (19x+3)*x=14 equation:


Simplifying
(19x + 3) * x = 14

Reorder the terms:
(3 + 19x) * x = 14

Reorder the terms for easier multiplication:
x(3 + 19x) = 14
(3 * x + 19x * x) = 14
(3x + 19x2) = 14

Solving
3x + 19x2 = 14

Solving for variable 'x'.

Reorder the terms:
-14 + 3x + 19x2 = 14 + -14

Combine like terms: 14 + -14 = 0
-14 + 3x + 19x2 = 0

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

Divide each side by '19'.
-0.7368421053 + 0.1578947368x + x2 = 0

Move the constant term to the right:

Add '0.7368421053' to each side of the equation.
-0.7368421053 + 0.1578947368x + 0.7368421053 + x2 = 0 + 0.7368421053

Reorder the terms:
-0.7368421053 + 0.7368421053 + 0.1578947368x + x2 = 0 + 0.7368421053

Combine like terms: -0.7368421053 + 0.7368421053 = 0.0000000000
0.0000000000 + 0.1578947368x + x2 = 0 + 0.7368421053
0.1578947368x + x2 = 0 + 0.7368421053

Combine like terms: 0 + 0.7368421053 = 0.7368421053
0.1578947368x + x2 = 0.7368421053

The x term is 0.1578947368x.  Take half its coefficient (0.0789473684).
Square it (0.006232686977) and add it to both sides.

Add '0.006232686977' to each side of the equation.
0.1578947368x + 0.006232686977 + x2 = 0.7368421053 + 0.006232686977

Reorder the terms:
0.006232686977 + 0.1578947368x + x2 = 0.7368421053 + 0.006232686977

Combine like terms: 0.7368421053 + 0.006232686977 = 0.743074792277
0.006232686977 + 0.1578947368x + x2 = 0.743074792277

Factor a perfect square on the left side:
(x + 0.0789473684)(x + 0.0789473684) = 0.743074792277

Calculate the square root of the right side: 0.862017861

Break this problem into two subproblems by setting 
(x + 0.0789473684) equal to 0.862017861 and -0.862017861.

Subproblem 1

x + 0.0789473684 = 0.862017861 Simplifying x + 0.0789473684 = 0.862017861 Reorder the terms: 0.0789473684 + x = 0.862017861 Solving 0.0789473684 + x = 0.862017861 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0789473684' to each side of the equation. 0.0789473684 + -0.0789473684 + x = 0.862017861 + -0.0789473684 Combine like terms: 0.0789473684 + -0.0789473684 = 0.0000000000 0.0000000000 + x = 0.862017861 + -0.0789473684 x = 0.862017861 + -0.0789473684 Combine like terms: 0.862017861 + -0.0789473684 = 0.7830704926 x = 0.7830704926 Simplifying x = 0.7830704926

Subproblem 2

x + 0.0789473684 = -0.862017861 Simplifying x + 0.0789473684 = -0.862017861 Reorder the terms: 0.0789473684 + x = -0.862017861 Solving 0.0789473684 + x = -0.862017861 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0789473684' to each side of the equation. 0.0789473684 + -0.0789473684 + x = -0.862017861 + -0.0789473684 Combine like terms: 0.0789473684 + -0.0789473684 = 0.0000000000 0.0000000000 + x = -0.862017861 + -0.0789473684 x = -0.862017861 + -0.0789473684 Combine like terms: -0.862017861 + -0.0789473684 = -0.9409652294 x = -0.9409652294 Simplifying x = -0.9409652294

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.7830704926, -0.9409652294}

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