7.69x^2+2.78x=5.37

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Solution for 7.69x^2+2.78x=5.37 equation:


Simplifying
7.69x2 + 2.78x = 5.37

Reorder the terms:
2.78x + 7.69x2 = 5.37

Solving
2.78x + 7.69x2 = 5.37

Solving for variable 'x'.

Reorder the terms:
-5.37 + 2.78x + 7.69x2 = 5.37 + -5.37

Combine like terms: 5.37 + -5.37 = 0.00
-5.37 + 2.78x + 7.69x2 = 0.00

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

Divide each side by '7.69'.
-0.6983094928 + 0.3615084525x + x2 = 0

Move the constant term to the right:

Add '0.6983094928' to each side of the equation.
-0.6983094928 + 0.3615084525x + 0.6983094928 + x2 = 0 + 0.6983094928

Reorder the terms:
-0.6983094928 + 0.6983094928 + 0.3615084525x + x2 = 0 + 0.6983094928

Combine like terms: -0.6983094928 + 0.6983094928 = 0.0000000000
0.0000000000 + 0.3615084525x + x2 = 0 + 0.6983094928
0.3615084525x + x2 = 0 + 0.6983094928

Combine like terms: 0 + 0.6983094928 = 0.6983094928
0.3615084525x + x2 = 0.6983094928

The x term is 0.3615084525x.  Take half its coefficient (0.1807542263).
Square it (0.03267209033) and add it to both sides.

Add '0.03267209033' to each side of the equation.
0.3615084525x + 0.03267209033 + x2 = 0.6983094928 + 0.03267209033

Reorder the terms:
0.03267209033 + 0.3615084525x + x2 = 0.6983094928 + 0.03267209033

Combine like terms: 0.6983094928 + 0.03267209033 = 0.73098158313
0.03267209033 + 0.3615084525x + x2 = 0.73098158313

Factor a perfect square on the left side:
(x + 0.1807542263)(x + 0.1807542263) = 0.73098158313

Calculate the square root of the right side: 0.85497461

Break this problem into two subproblems by setting 
(x + 0.1807542263) equal to 0.85497461 and -0.85497461.

Subproblem 1

x + 0.1807542263 = 0.85497461 Simplifying x + 0.1807542263 = 0.85497461 Reorder the terms: 0.1807542263 + x = 0.85497461 Solving 0.1807542263 + x = 0.85497461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1807542263' to each side of the equation. 0.1807542263 + -0.1807542263 + x = 0.85497461 + -0.1807542263 Combine like terms: 0.1807542263 + -0.1807542263 = 0.0000000000 0.0000000000 + x = 0.85497461 + -0.1807542263 x = 0.85497461 + -0.1807542263 Combine like terms: 0.85497461 + -0.1807542263 = 0.6742203837 x = 0.6742203837 Simplifying x = 0.6742203837

Subproblem 2

x + 0.1807542263 = -0.85497461 Simplifying x + 0.1807542263 = -0.85497461 Reorder the terms: 0.1807542263 + x = -0.85497461 Solving 0.1807542263 + x = -0.85497461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1807542263' to each side of the equation. 0.1807542263 + -0.1807542263 + x = -0.85497461 + -0.1807542263 Combine like terms: 0.1807542263 + -0.1807542263 = 0.0000000000 0.0000000000 + x = -0.85497461 + -0.1807542263 x = -0.85497461 + -0.1807542263 Combine like terms: -0.85497461 + -0.1807542263 = -1.0357288363 x = -1.0357288363 Simplifying x = -1.0357288363

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.6742203837, -1.0357288363}

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