7x^2=6-5x

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Solution for 7x^2=6-5x equation:


Simplifying
7x2 = 6 + -5x

Solving
7x2 = 6 + -5x

Solving for variable 'x'.

Reorder the terms:
-6 + 5x + 7x2 = 6 + -5x + -6 + 5x

Reorder the terms:
-6 + 5x + 7x2 = 6 + -6 + -5x + 5x

Combine like terms: 6 + -6 = 0
-6 + 5x + 7x2 = 0 + -5x + 5x
-6 + 5x + 7x2 = -5x + 5x

Combine like terms: -5x + 5x = 0
-6 + 5x + 7x2 = 0

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

Divide each side by '7'.
-0.8571428571 + 0.7142857143x + x2 = 0

Move the constant term to the right:

Add '0.8571428571' to each side of the equation.
-0.8571428571 + 0.7142857143x + 0.8571428571 + x2 = 0 + 0.8571428571

Reorder the terms:
-0.8571428571 + 0.8571428571 + 0.7142857143x + x2 = 0 + 0.8571428571

Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000
0.0000000000 + 0.7142857143x + x2 = 0 + 0.8571428571
0.7142857143x + x2 = 0 + 0.8571428571

Combine like terms: 0 + 0.8571428571 = 0.8571428571
0.7142857143x + x2 = 0.8571428571

The x term is 0.7142857143x.  Take half its coefficient (0.3571428572).
Square it (0.1275510204) and add it to both sides.

Add '0.1275510204' to each side of the equation.
0.7142857143x + 0.1275510204 + x2 = 0.8571428571 + 0.1275510204

Reorder the terms:
0.1275510204 + 0.7142857143x + x2 = 0.8571428571 + 0.1275510204

Combine like terms: 0.8571428571 + 0.1275510204 = 0.9846938775
0.1275510204 + 0.7142857143x + x2 = 0.9846938775

Factor a perfect square on the left side:
(x + 0.3571428572)(x + 0.3571428572) = 0.9846938775

Calculate the square root of the right side: 0.992317428

Break this problem into two subproblems by setting 
(x + 0.3571428572) equal to 0.992317428 and -0.992317428.

Subproblem 1

x + 0.3571428572 = 0.992317428 Simplifying x + 0.3571428572 = 0.992317428 Reorder the terms: 0.3571428572 + x = 0.992317428 Solving 0.3571428572 + x = 0.992317428 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3571428572' to each side of the equation. 0.3571428572 + -0.3571428572 + x = 0.992317428 + -0.3571428572 Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000 0.0000000000 + x = 0.992317428 + -0.3571428572 x = 0.992317428 + -0.3571428572 Combine like terms: 0.992317428 + -0.3571428572 = 0.6351745708 x = 0.6351745708 Simplifying x = 0.6351745708

Subproblem 2

x + 0.3571428572 = -0.992317428 Simplifying x + 0.3571428572 = -0.992317428 Reorder the terms: 0.3571428572 + x = -0.992317428 Solving 0.3571428572 + x = -0.992317428 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3571428572' to each side of the equation. 0.3571428572 + -0.3571428572 + x = -0.992317428 + -0.3571428572 Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000 0.0000000000 + x = -0.992317428 + -0.3571428572 x = -0.992317428 + -0.3571428572 Combine like terms: -0.992317428 + -0.3571428572 = -1.3494602852 x = -1.3494602852 Simplifying x = -1.3494602852

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.6351745708, -1.3494602852}

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