6x^2-21x-7=0

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Solution for 6x^2-21x-7=0 equation:


Simplifying
6x2 + -21x + -7 = 0

Reorder the terms:
-7 + -21x + 6x2 = 0

Solving
-7 + -21x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-1.166666667 + -3.5x + x2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + -3.5x + 1.166666667 + x2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + -3.5x + x2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + -3.5x + x2 = 0 + 1.166666667
-3.5x + x2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
-3.5x + x2 = 1.166666667

The x term is -3.5x.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5x + 3.0625 + x2 = 1.166666667 + 3.0625

Reorder the terms:
3.0625 + -3.5x + x2 = 1.166666667 + 3.0625

Combine like terms: 1.166666667 + 3.0625 = 4.229166667
3.0625 + -3.5x + x2 = 4.229166667

Factor a perfect square on the left side:
(x + -1.75)(x + -1.75) = 4.229166667

Calculate the square root of the right side: 2.05649378

Break this problem into two subproblems by setting 
(x + -1.75) equal to 2.05649378 and -2.05649378.

Subproblem 1

x + -1.75 = 2.05649378 Simplifying x + -1.75 = 2.05649378 Reorder the terms: -1.75 + x = 2.05649378 Solving -1.75 + x = 2.05649378 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x = 2.05649378 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x = 2.05649378 + 1.75 x = 2.05649378 + 1.75 Combine like terms: 2.05649378 + 1.75 = 3.80649378 x = 3.80649378 Simplifying x = 3.80649378

Subproblem 2

x + -1.75 = -2.05649378 Simplifying x + -1.75 = -2.05649378 Reorder the terms: -1.75 + x = -2.05649378 Solving -1.75 + x = -2.05649378 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x = -2.05649378 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x = -2.05649378 + 1.75 x = -2.05649378 + 1.75 Combine like terms: -2.05649378 + 1.75 = -0.30649378 x = -0.30649378 Simplifying x = -0.30649378

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.80649378, -0.30649378}

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