7x^2+22x-7=0

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Solution for 7x^2+22x-7=0 equation:


Simplifying
7x2 + 22x + -7 = 0

Reorder the terms:
-7 + 22x + 7x2 = 0

Solving
-7 + 22x + 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'.
-1 + 3.142857143x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 3.142857143x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 3.142857143x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 3.142857143x + x2 = 0 + 1
3.142857143x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
3.142857143x + x2 = 1

The x term is 3.142857143x.  Take half its coefficient (1.571428572).
Square it (2.469387757) and add it to both sides.

Add '2.469387757' to each side of the equation.
3.142857143x + 2.469387757 + x2 = 1 + 2.469387757

Reorder the terms:
2.469387757 + 3.142857143x + x2 = 1 + 2.469387757

Combine like terms: 1 + 2.469387757 = 3.469387757
2.469387757 + 3.142857143x + x2 = 3.469387757

Factor a perfect square on the left side:
(x + 1.571428572)(x + 1.571428572) = 3.469387757

Calculate the square root of the right side: 1.862629259

Break this problem into two subproblems by setting 
(x + 1.571428572) equal to 1.862629259 and -1.862629259.

Subproblem 1

x + 1.571428572 = 1.862629259 Simplifying x + 1.571428572 = 1.862629259 Reorder the terms: 1.571428572 + x = 1.862629259 Solving 1.571428572 + x = 1.862629259 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.571428572' to each side of the equation. 1.571428572 + -1.571428572 + x = 1.862629259 + -1.571428572 Combine like terms: 1.571428572 + -1.571428572 = 0.000000000 0.000000000 + x = 1.862629259 + -1.571428572 x = 1.862629259 + -1.571428572 Combine like terms: 1.862629259 + -1.571428572 = 0.291200687 x = 0.291200687 Simplifying x = 0.291200687

Subproblem 2

x + 1.571428572 = -1.862629259 Simplifying x + 1.571428572 = -1.862629259 Reorder the terms: 1.571428572 + x = -1.862629259 Solving 1.571428572 + x = -1.862629259 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.571428572' to each side of the equation. 1.571428572 + -1.571428572 + x = -1.862629259 + -1.571428572 Combine like terms: 1.571428572 + -1.571428572 = 0.000000000 0.000000000 + x = -1.862629259 + -1.571428572 x = -1.862629259 + -1.571428572 Combine like terms: -1.862629259 + -1.571428572 = -3.434057831 x = -3.434057831 Simplifying x = -3.434057831

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.291200687, -3.434057831}

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