x^2+6x=78

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Solution for x^2+6x=78 equation:


Simplifying
x2 + 6x = 78

Reorder the terms:
6x + x2 = 78

Solving
6x + x2 = 78

Solving for variable 'x'.

Reorder the terms:
-78 + 6x + x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-78 + 6x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + 6x + 78 + x2 = 0 + 78

Reorder the terms:
-78 + 78 + 6x + x2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + 6x + x2 = 0 + 78
6x + x2 = 0 + 78

Combine like terms: 0 + 78 = 78
6x + x2 = 78

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = 78 + 9

Reorder the terms:
9 + 6x + x2 = 78 + 9

Combine like terms: 78 + 9 = 87
9 + 6x + x2 = 87

Factor a perfect square on the left side:
(x + 3)(x + 3) = 87

Calculate the square root of the right side: 9.327379053

Break this problem into two subproblems by setting 
(x + 3) equal to 9.327379053 and -9.327379053.

Subproblem 1

x + 3 = 9.327379053 Simplifying x + 3 = 9.327379053 Reorder the terms: 3 + x = 9.327379053 Solving 3 + x = 9.327379053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 9.327379053 + -3 Combine like terms: 3 + -3 = 0 0 + x = 9.327379053 + -3 x = 9.327379053 + -3 Combine like terms: 9.327379053 + -3 = 6.327379053 x = 6.327379053 Simplifying x = 6.327379053

Subproblem 2

x + 3 = -9.327379053 Simplifying x + 3 = -9.327379053 Reorder the terms: 3 + x = -9.327379053 Solving 3 + x = -9.327379053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = -9.327379053 + -3 Combine like terms: 3 + -3 = 0 0 + x = -9.327379053 + -3 x = -9.327379053 + -3 Combine like terms: -9.327379053 + -3 = -12.327379053 x = -12.327379053 Simplifying x = -12.327379053

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.327379053, -12.327379053}

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