14x^2+31x-15=0

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Solution for 14x^2+31x-15=0 equation:


Simplifying
14x2 + 31x + -15 = 0

Reorder the terms:
-15 + 31x + 14x2 = 0

Solving
-15 + 31x + 14x2 = 0

Solving for variable 'x'.

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

Divide each side by '14'.
-1.071428571 + 2.214285714x + x2 = 0

Move the constant term to the right:

Add '1.071428571' to each side of the equation.
-1.071428571 + 2.214285714x + 1.071428571 + x2 = 0 + 1.071428571

Reorder the terms:
-1.071428571 + 1.071428571 + 2.214285714x + x2 = 0 + 1.071428571

Combine like terms: -1.071428571 + 1.071428571 = 0.000000000
0.000000000 + 2.214285714x + x2 = 0 + 1.071428571
2.214285714x + x2 = 0 + 1.071428571

Combine like terms: 0 + 1.071428571 = 1.071428571
2.214285714x + x2 = 1.071428571

The x term is 2.214285714x.  Take half its coefficient (1.107142857).
Square it (1.225765306) and add it to both sides.

Add '1.225765306' to each side of the equation.
2.214285714x + 1.225765306 + x2 = 1.071428571 + 1.225765306

Reorder the terms:
1.225765306 + 2.214285714x + x2 = 1.071428571 + 1.225765306

Combine like terms: 1.071428571 + 1.225765306 = 2.297193877
1.225765306 + 2.214285714x + x2 = 2.297193877

Factor a perfect square on the left side:
(x + 1.107142857)(x + 1.107142857) = 2.297193877

Calculate the square root of the right side: 1.515649655

Break this problem into two subproblems by setting 
(x + 1.107142857) equal to 1.515649655 and -1.515649655.

Subproblem 1

x + 1.107142857 = 1.515649655 Simplifying x + 1.107142857 = 1.515649655 Reorder the terms: 1.107142857 + x = 1.515649655 Solving 1.107142857 + x = 1.515649655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.107142857' to each side of the equation. 1.107142857 + -1.107142857 + x = 1.515649655 + -1.107142857 Combine like terms: 1.107142857 + -1.107142857 = 0.000000000 0.000000000 + x = 1.515649655 + -1.107142857 x = 1.515649655 + -1.107142857 Combine like terms: 1.515649655 + -1.107142857 = 0.408506798 x = 0.408506798 Simplifying x = 0.408506798

Subproblem 2

x + 1.107142857 = -1.515649655 Simplifying x + 1.107142857 = -1.515649655 Reorder the terms: 1.107142857 + x = -1.515649655 Solving 1.107142857 + x = -1.515649655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.107142857' to each side of the equation. 1.107142857 + -1.107142857 + x = -1.515649655 + -1.107142857 Combine like terms: 1.107142857 + -1.107142857 = 0.000000000 0.000000000 + x = -1.515649655 + -1.107142857 x = -1.515649655 + -1.107142857 Combine like terms: -1.515649655 + -1.107142857 = -2.622792512 x = -2.622792512 Simplifying x = -2.622792512

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.408506798, -2.622792512}

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