7x^2+41x-14=0

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Solution for 7x^2+41x-14=0 equation:


Simplifying
7x2 + 41x + -14 = 0

Reorder the terms:
-14 + 41x + 7x2 = 0

Solving
-14 + 41x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
-2 + 5.857142857x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 5.857142857x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + 5.857142857x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 5.857142857x + x2 = 0 + 2
5.857142857x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
5.857142857x + x2 = 2

The x term is 5.857142857x.  Take half its coefficient (2.928571429).
Square it (8.576530615) and add it to both sides.

Add '8.576530615' to each side of the equation.
5.857142857x + 8.576530615 + x2 = 2 + 8.576530615

Reorder the terms:
8.576530615 + 5.857142857x + x2 = 2 + 8.576530615

Combine like terms: 2 + 8.576530615 = 10.576530615
8.576530615 + 5.857142857x + x2 = 10.576530615

Factor a perfect square on the left side:
(x + 2.928571429)(x + 2.928571429) = 10.576530615

Calculate the square root of the right side: 3.25215784

Break this problem into two subproblems by setting 
(x + 2.928571429) equal to 3.25215784 and -3.25215784.

Subproblem 1

x + 2.928571429 = 3.25215784 Simplifying x + 2.928571429 = 3.25215784 Reorder the terms: 2.928571429 + x = 3.25215784 Solving 2.928571429 + x = 3.25215784 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.928571429' to each side of the equation. 2.928571429 + -2.928571429 + x = 3.25215784 + -2.928571429 Combine like terms: 2.928571429 + -2.928571429 = 0.000000000 0.000000000 + x = 3.25215784 + -2.928571429 x = 3.25215784 + -2.928571429 Combine like terms: 3.25215784 + -2.928571429 = 0.323586411 x = 0.323586411 Simplifying x = 0.323586411

Subproblem 2

x + 2.928571429 = -3.25215784 Simplifying x + 2.928571429 = -3.25215784 Reorder the terms: 2.928571429 + x = -3.25215784 Solving 2.928571429 + x = -3.25215784 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.928571429' to each side of the equation. 2.928571429 + -2.928571429 + x = -3.25215784 + -2.928571429 Combine like terms: 2.928571429 + -2.928571429 = 0.000000000 0.000000000 + x = -3.25215784 + -2.928571429 x = -3.25215784 + -2.928571429 Combine like terms: -3.25215784 + -2.928571429 = -6.180729269 x = -6.180729269 Simplifying x = -6.180729269

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.323586411, -6.180729269}

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