13x^2-14x=9

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Solution for 13x^2-14x=9 equation:


Simplifying
13x2 + -14x = 9

Reorder the terms:
-14x + 13x2 = 9

Solving
-14x + 13x2 = 9

Solving for variable 'x'.

Reorder the terms:
-9 + -14x + 13x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + -14x + 13x2 = 0

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

Divide each side by '13'.
-0.6923076923 + -1.076923077x + x2 = 0

Move the constant term to the right:

Add '0.6923076923' to each side of the equation.
-0.6923076923 + -1.076923077x + 0.6923076923 + x2 = 0 + 0.6923076923

Reorder the terms:
-0.6923076923 + 0.6923076923 + -1.076923077x + x2 = 0 + 0.6923076923

Combine like terms: -0.6923076923 + 0.6923076923 = 0.0000000000
0.0000000000 + -1.076923077x + x2 = 0 + 0.6923076923
-1.076923077x + x2 = 0 + 0.6923076923

Combine like terms: 0 + 0.6923076923 = 0.6923076923
-1.076923077x + x2 = 0.6923076923

The x term is -1.076923077x.  Take half its coefficient (-0.5384615385).
Square it (0.2899408284) and add it to both sides.

Add '0.2899408284' to each side of the equation.
-1.076923077x + 0.2899408284 + x2 = 0.6923076923 + 0.2899408284

Reorder the terms:
0.2899408284 + -1.076923077x + x2 = 0.6923076923 + 0.2899408284

Combine like terms: 0.6923076923 + 0.2899408284 = 0.9822485207
0.2899408284 + -1.076923077x + x2 = 0.9822485207

Factor a perfect square on the left side:
(x + -0.5384615385)(x + -0.5384615385) = 0.9822485207

Calculate the square root of the right side: 0.991084517

Break this problem into two subproblems by setting 
(x + -0.5384615385) equal to 0.991084517 and -0.991084517.

Subproblem 1

x + -0.5384615385 = 0.991084517 Simplifying x + -0.5384615385 = 0.991084517 Reorder the terms: -0.5384615385 + x = 0.991084517 Solving -0.5384615385 + x = 0.991084517 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5384615385' to each side of the equation. -0.5384615385 + 0.5384615385 + x = 0.991084517 + 0.5384615385 Combine like terms: -0.5384615385 + 0.5384615385 = 0.0000000000 0.0000000000 + x = 0.991084517 + 0.5384615385 x = 0.991084517 + 0.5384615385 Combine like terms: 0.991084517 + 0.5384615385 = 1.5295460555 x = 1.5295460555 Simplifying x = 1.5295460555

Subproblem 2

x + -0.5384615385 = -0.991084517 Simplifying x + -0.5384615385 = -0.991084517 Reorder the terms: -0.5384615385 + x = -0.991084517 Solving -0.5384615385 + x = -0.991084517 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5384615385' to each side of the equation. -0.5384615385 + 0.5384615385 + x = -0.991084517 + 0.5384615385 Combine like terms: -0.5384615385 + 0.5384615385 = 0.0000000000 0.0000000000 + x = -0.991084517 + 0.5384615385 x = -0.991084517 + 0.5384615385 Combine like terms: -0.991084517 + 0.5384615385 = -0.4526229785 x = -0.4526229785 Simplifying x = -0.4526229785

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.5295460555, -0.4526229785}

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