6x^2+5x=9

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Solution for 6x^2+5x=9 equation:


Simplifying
6x2 + 5x = 9

Reorder the terms:
5x + 6x2 = 9

Solving
5x + 6x2 = 9

Solving for variable 'x'.

Reorder the terms:
-9 + 5x + 6x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 5x + 6x2 = 0

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

Divide each side by '6'.
-1.5 + 0.8333333333x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 0.8333333333x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 0.8333333333x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 0.8333333333x + x2 = 0 + 1.5
0.8333333333x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
0.8333333333x + x2 = 1.5

The x term is 0.8333333333x.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333x + 0.1736111111 + x2 = 1.5 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333x + x2 = 1.5 + 0.1736111111

Combine like terms: 1.5 + 0.1736111111 = 1.6736111111
0.1736111111 + 0.8333333333x + x2 = 1.6736111111

Factor a perfect square on the left side:
(x + 0.4166666667)(x + 0.4166666667) = 1.6736111111

Calculate the square root of the right side: 1.293681225

Break this problem into two subproblems by setting 
(x + 0.4166666667) equal to 1.293681225 and -1.293681225.

Subproblem 1

x + 0.4166666667 = 1.293681225 Simplifying x + 0.4166666667 = 1.293681225 Reorder the terms: 0.4166666667 + x = 1.293681225 Solving 0.4166666667 + x = 1.293681225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = 1.293681225 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 1.293681225 + -0.4166666667 x = 1.293681225 + -0.4166666667 Combine like terms: 1.293681225 + -0.4166666667 = 0.8770145583 x = 0.8770145583 Simplifying x = 0.8770145583

Subproblem 2

x + 0.4166666667 = -1.293681225 Simplifying x + 0.4166666667 = -1.293681225 Reorder the terms: 0.4166666667 + x = -1.293681225 Solving 0.4166666667 + x = -1.293681225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = -1.293681225 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -1.293681225 + -0.4166666667 x = -1.293681225 + -0.4166666667 Combine like terms: -1.293681225 + -0.4166666667 = -1.7103478917 x = -1.7103478917 Simplifying x = -1.7103478917

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.8770145583, -1.7103478917}

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