4.2x^2+6.5x-6.7=0

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Solution for 4.2x^2+6.5x-6.7=0 equation:


Simplifying
4.2x2 + 6.5x + -6.7 = 0

Reorder the terms:
-6.7 + 6.5x + 4.2x2 = 0

Solving
-6.7 + 6.5x + 4.2x2 = 0

Solving for variable 'x'.

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

Divide each side by '4.2'.
-1.595238095 + 1.547619048x + x2 = 0

Move the constant term to the right:

Add '1.595238095' to each side of the equation.
-1.595238095 + 1.547619048x + 1.595238095 + x2 = 0 + 1.595238095

Reorder the terms:
-1.595238095 + 1.595238095 + 1.547619048x + x2 = 0 + 1.595238095

Combine like terms: -1.595238095 + 1.595238095 = 0.000000000
0.000000000 + 1.547619048x + x2 = 0 + 1.595238095
1.547619048x + x2 = 0 + 1.595238095

Combine like terms: 0 + 1.595238095 = 1.595238095
1.547619048x + x2 = 1.595238095

The x term is 1.547619048x.  Take half its coefficient (0.773809524).
Square it (0.5987811794) and add it to both sides.

Add '0.5987811794' to each side of the equation.
1.547619048x + 0.5987811794 + x2 = 1.595238095 + 0.5987811794

Reorder the terms:
0.5987811794 + 1.547619048x + x2 = 1.595238095 + 0.5987811794

Combine like terms: 1.595238095 + 0.5987811794 = 2.1940192744
0.5987811794 + 1.547619048x + x2 = 2.1940192744

Factor a perfect square on the left side:
(x + 0.773809524)(x + 0.773809524) = 2.1940192744

Calculate the square root of the right side: 1.481222223

Break this problem into two subproblems by setting 
(x + 0.773809524) equal to 1.481222223 and -1.481222223.

Subproblem 1

x + 0.773809524 = 1.481222223 Simplifying x + 0.773809524 = 1.481222223 Reorder the terms: 0.773809524 + x = 1.481222223 Solving 0.773809524 + x = 1.481222223 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.773809524' to each side of the equation. 0.773809524 + -0.773809524 + x = 1.481222223 + -0.773809524 Combine like terms: 0.773809524 + -0.773809524 = 0.000000000 0.000000000 + x = 1.481222223 + -0.773809524 x = 1.481222223 + -0.773809524 Combine like terms: 1.481222223 + -0.773809524 = 0.707412699 x = 0.707412699 Simplifying x = 0.707412699

Subproblem 2

x + 0.773809524 = -1.481222223 Simplifying x + 0.773809524 = -1.481222223 Reorder the terms: 0.773809524 + x = -1.481222223 Solving 0.773809524 + x = -1.481222223 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.773809524' to each side of the equation. 0.773809524 + -0.773809524 + x = -1.481222223 + -0.773809524 Combine like terms: 0.773809524 + -0.773809524 = 0.000000000 0.000000000 + x = -1.481222223 + -0.773809524 x = -1.481222223 + -0.773809524 Combine like terms: -1.481222223 + -0.773809524 = -2.255031747 x = -2.255031747 Simplifying x = -2.255031747

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.707412699, -2.255031747}

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