6x^2+7x-21=0

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


Simplifying
6x2 + 7x + -21 = 0

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

Solving
-21 + 7x + 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'.
-3.5 + 1.166666667x + x2 = 0

Move the constant term to the right:

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

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

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 1.166666667x + x2 = 0 + 3.5
1.166666667x + x2 = 0 + 3.5

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

The x term is 1.166666667x.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667x + 0.3402777780 + x2 = 3.5 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667x + x2 = 3.5 + 0.3402777780

Combine like terms: 3.5 + 0.3402777780 = 3.840277778
0.3402777780 + 1.166666667x + x2 = 3.840277778

Factor a perfect square on the left side:
(x + 0.5833333335)(x + 0.5833333335) = 3.840277778

Calculate the square root of the right side: 1.959662669

Break this problem into two subproblems by setting 
(x + 0.5833333335) equal to 1.959662669 and -1.959662669.

Subproblem 1

x + 0.5833333335 = 1.959662669 Simplifying x + 0.5833333335 = 1.959662669 Reorder the terms: 0.5833333335 + x = 1.959662669 Solving 0.5833333335 + x = 1.959662669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = 1.959662669 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = 1.959662669 + -0.5833333335 x = 1.959662669 + -0.5833333335 Combine like terms: 1.959662669 + -0.5833333335 = 1.3763293355 x = 1.3763293355 Simplifying x = 1.3763293355

Subproblem 2

x + 0.5833333335 = -1.959662669 Simplifying x + 0.5833333335 = -1.959662669 Reorder the terms: 0.5833333335 + x = -1.959662669 Solving 0.5833333335 + x = -1.959662669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = -1.959662669 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = -1.959662669 + -0.5833333335 x = -1.959662669 + -0.5833333335 Combine like terms: -1.959662669 + -0.5833333335 = -2.5429960025 x = -2.5429960025 Simplifying x = -2.5429960025

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3763293355, -2.5429960025}

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