6x^2+23x-378=0

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Solution for 6x^2+23x-378=0 equation:


Simplifying
6x2 + 23x + -378 = 0

Reorder the terms:
-378 + 23x + 6x2 = 0

Solving
-378 + 23x + 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'.
-63 + 3.833333333x + x2 = 0

Move the constant term to the right:

Add '63' to each side of the equation.
-63 + 3.833333333x + 63 + x2 = 0 + 63

Reorder the terms:
-63 + 63 + 3.833333333x + x2 = 0 + 63

Combine like terms: -63 + 63 = 0
0 + 3.833333333x + x2 = 0 + 63
3.833333333x + x2 = 0 + 63

Combine like terms: 0 + 63 = 63
3.833333333x + x2 = 63

The x term is 3.833333333x.  Take half its coefficient (1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
3.833333333x + 3.673611112 + x2 = 63 + 3.673611112

Reorder the terms:
3.673611112 + 3.833333333x + x2 = 63 + 3.673611112

Combine like terms: 63 + 3.673611112 = 66.673611112
3.673611112 + 3.833333333x + x2 = 66.673611112

Factor a perfect square on the left side:
(x + 1.916666667)(x + 1.916666667) = 66.673611112

Calculate the square root of the right side: 8.165391057

Break this problem into two subproblems by setting 
(x + 1.916666667) equal to 8.165391057 and -8.165391057.

Subproblem 1

x + 1.916666667 = 8.165391057 Simplifying x + 1.916666667 = 8.165391057 Reorder the terms: 1.916666667 + x = 8.165391057 Solving 1.916666667 + x = 8.165391057 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = 8.165391057 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = 8.165391057 + -1.916666667 x = 8.165391057 + -1.916666667 Combine like terms: 8.165391057 + -1.916666667 = 6.24872439 x = 6.24872439 Simplifying x = 6.24872439

Subproblem 2

x + 1.916666667 = -8.165391057 Simplifying x + 1.916666667 = -8.165391057 Reorder the terms: 1.916666667 + x = -8.165391057 Solving 1.916666667 + x = -8.165391057 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = -8.165391057 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = -8.165391057 + -1.916666667 x = -8.165391057 + -1.916666667 Combine like terms: -8.165391057 + -1.916666667 = -10.082057724 x = -10.082057724 Simplifying x = -10.082057724

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.24872439, -10.082057724}

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