3x^2+x-114=78

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Solution for 3x^2+x-114=78 equation:


Simplifying
3x2 + x + -114 = 78

Reorder the terms:
-114 + x + 3x2 = 78

Solving
-114 + x + 3x2 = 78

Solving for variable 'x'.

Reorder the terms:
-114 + -78 + x + 3x2 = 78 + -78

Combine like terms: -114 + -78 = -192
-192 + x + 3x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-192 + x + 3x2 = 0

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

Divide each side by '3'.
-64 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '64' to each side of the equation.
-64 + 0.3333333333x + 64 + x2 = 0 + 64

Reorder the terms:
-64 + 64 + 0.3333333333x + x2 = 0 + 64

Combine like terms: -64 + 64 = 0
0 + 0.3333333333x + x2 = 0 + 64
0.3333333333x + x2 = 0 + 64

Combine like terms: 0 + 64 = 64
0.3333333333x + x2 = 64

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.3333333333x + 0.25 + x2 = 64 + 0.25

Reorder the terms:
0.25 + 0.3333333333x + x2 = 64 + 0.25

Combine like terms: 64 + 0.25 = 64.25
0.25 + 0.3333333333x + x2 = 64.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 64.25

Calculate the square root of the right side: 8.015609771

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.015609771 and -8.015609771.

Subproblem 1

x + 0.5 = 8.015609771 Simplifying x + 0.5 = 8.015609771 Reorder the terms: 0.5 + x = 8.015609771 Solving 0.5 + x = 8.015609771 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 8.015609771 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.015609771 + -0.5 x = 8.015609771 + -0.5 Combine like terms: 8.015609771 + -0.5 = 7.515609771 x = 7.515609771 Simplifying x = 7.515609771

Subproblem 2

x + 0.5 = -8.015609771 Simplifying x + 0.5 = -8.015609771 Reorder the terms: 0.5 + x = -8.015609771 Solving 0.5 + x = -8.015609771 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -8.015609771 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.015609771 + -0.5 x = -8.015609771 + -0.5 Combine like terms: -8.015609771 + -0.5 = -8.515609771 x = -8.515609771 Simplifying x = -8.515609771

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.515609771, -8.515609771}

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