x^2+2x-78=5

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Solution for x^2+2x-78=5 equation:


Simplifying
x2 + 2x + -78 = 5

Reorder the terms:
-78 + 2x + x2 = 5

Solving
-78 + 2x + x2 = 5

Solving for variable 'x'.

Reorder the terms:
-78 + -5 + 2x + x2 = 5 + -5

Combine like terms: -78 + -5 = -83
-83 + 2x + x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-83 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '83' to each side of the equation.
-83 + 2x + 83 + x2 = 0 + 83

Reorder the terms:
-83 + 83 + 2x + x2 = 0 + 83

Combine like terms: -83 + 83 = 0
0 + 2x + x2 = 0 + 83
2x + x2 = 0 + 83

Combine like terms: 0 + 83 = 83
2x + x2 = 83

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 83 + 1

Reorder the terms:
1 + 2x + x2 = 83 + 1

Combine like terms: 83 + 1 = 84
1 + 2x + x2 = 84

Factor a perfect square on the left side:
(x + 1)(x + 1) = 84

Calculate the square root of the right side: 9.16515139

Break this problem into two subproblems by setting 
(x + 1) equal to 9.16515139 and -9.16515139.

Subproblem 1

x + 1 = 9.16515139 Simplifying x + 1 = 9.16515139 Reorder the terms: 1 + x = 9.16515139 Solving 1 + x = 9.16515139 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 9.16515139 + -1 Combine like terms: 1 + -1 = 0 0 + x = 9.16515139 + -1 x = 9.16515139 + -1 Combine like terms: 9.16515139 + -1 = 8.16515139 x = 8.16515139 Simplifying x = 8.16515139

Subproblem 2

x + 1 = -9.16515139 Simplifying x + 1 = -9.16515139 Reorder the terms: 1 + x = -9.16515139 Solving 1 + x = -9.16515139 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -9.16515139 + -1 Combine like terms: 1 + -1 = 0 0 + x = -9.16515139 + -1 x = -9.16515139 + -1 Combine like terms: -9.16515139 + -1 = -10.16515139 x = -10.16515139 Simplifying x = -10.16515139

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.16515139, -10.16515139}

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