21x^2-35x-54=0

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Solution for 21x^2-35x-54=0 equation:


Simplifying
21x2 + -35x + -54 = 0

Reorder the terms:
-54 + -35x + 21x2 = 0

Solving
-54 + -35x + 21x2 = 0

Solving for variable 'x'.

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

Divide each side by '21'.
-2.571428571 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '2.571428571' to each side of the equation.
-2.571428571 + -1.666666667x + 2.571428571 + x2 = 0 + 2.571428571

Reorder the terms:
-2.571428571 + 2.571428571 + -1.666666667x + x2 = 0 + 2.571428571

Combine like terms: -2.571428571 + 2.571428571 = 0.000000000
0.000000000 + -1.666666667x + x2 = 0 + 2.571428571
-1.666666667x + x2 = 0 + 2.571428571

Combine like terms: 0 + 2.571428571 = 2.571428571
-1.666666667x + x2 = 2.571428571

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 2.571428571 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 2.571428571 + 0.6944444447

Combine like terms: 2.571428571 + 0.6944444447 = 3.2658730157
0.6944444447 + -1.666666667x + x2 = 3.2658730157

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 3.2658730157

Calculate the square root of the right side: 1.807172658

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 1.807172658 and -1.807172658.

Subproblem 1

x + -0.8333333335 = 1.807172658 Simplifying x + -0.8333333335 = 1.807172658 Reorder the terms: -0.8333333335 + x = 1.807172658 Solving -0.8333333335 + x = 1.807172658 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 1.807172658 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 1.807172658 + 0.8333333335 x = 1.807172658 + 0.8333333335 Combine like terms: 1.807172658 + 0.8333333335 = 2.6405059915 x = 2.6405059915 Simplifying x = 2.6405059915

Subproblem 2

x + -0.8333333335 = -1.807172658 Simplifying x + -0.8333333335 = -1.807172658 Reorder the terms: -0.8333333335 + x = -1.807172658 Solving -0.8333333335 + x = -1.807172658 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -1.807172658 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -1.807172658 + 0.8333333335 x = -1.807172658 + 0.8333333335 Combine like terms: -1.807172658 + 0.8333333335 = -0.9738393245 x = -0.9738393245 Simplifying x = -0.9738393245

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.6405059915, -0.9738393245}

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