x(2+x)+x+2=78

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Solution for x(2+x)+x+2=78 equation:


Simplifying
x(2 + x) + x + 2 = 78
(2 * x + x * x) + x + 2 = 78
(2x + x2) + x + 2 = 78

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

Combine like terms: 2x + x = 3x
2 + 3x + x2 = 78

Solving
2 + 3x + x2 = 78

Solving for variable 'x'.

Reorder the terms:
2 + -78 + 3x + x2 = 78 + -78

Combine like terms: 2 + -78 = -76
-76 + 3x + x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-76 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '76' to each side of the equation.
-76 + 3x + 76 + x2 = 0 + 76

Reorder the terms:
-76 + 76 + 3x + x2 = 0 + 76

Combine like terms: -76 + 76 = 0
0 + 3x + x2 = 0 + 76
3x + x2 = 0 + 76

Combine like terms: 0 + 76 = 76
3x + x2 = 76

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 76 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 76 + 2.25

Combine like terms: 76 + 2.25 = 78.25
2.25 + 3x + x2 = 78.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 78.25

Calculate the square root of the right side: 8.845903006

Break this problem into two subproblems by setting 
(x + 1.5) equal to 8.845903006 and -8.845903006.

Subproblem 1

x + 1.5 = 8.845903006 Simplifying x + 1.5 = 8.845903006 Reorder the terms: 1.5 + x = 8.845903006 Solving 1.5 + x = 8.845903006 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 8.845903006 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 8.845903006 + -1.5 x = 8.845903006 + -1.5 Combine like terms: 8.845903006 + -1.5 = 7.345903006 x = 7.345903006 Simplifying x = 7.345903006

Subproblem 2

x + 1.5 = -8.845903006 Simplifying x + 1.5 = -8.845903006 Reorder the terms: 1.5 + x = -8.845903006 Solving 1.5 + x = -8.845903006 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -8.845903006 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -8.845903006 + -1.5 x = -8.845903006 + -1.5 Combine like terms: -8.845903006 + -1.5 = -10.345903006 x = -10.345903006 Simplifying x = -10.345903006

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.345903006, -10.345903006}

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