x^2+24x+23=78

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Solution for x^2+24x+23=78 equation:


Simplifying
x2 + 24x + 23 = 78

Reorder the terms:
23 + 24x + x2 = 78

Solving
23 + 24x + x2 = 78

Solving for variable 'x'.

Reorder the terms:
23 + -78 + 24x + x2 = 78 + -78

Combine like terms: 23 + -78 = -55
-55 + 24x + x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-55 + 24x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '55' to each side of the equation.
-55 + 24x + 55 + x2 = 0 + 55

Reorder the terms:
-55 + 55 + 24x + x2 = 0 + 55

Combine like terms: -55 + 55 = 0
0 + 24x + x2 = 0 + 55
24x + x2 = 0 + 55

Combine like terms: 0 + 55 = 55
24x + x2 = 55

The x term is 24x.  Take half its coefficient (12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
24x + 144 + x2 = 55 + 144

Reorder the terms:
144 + 24x + x2 = 55 + 144

Combine like terms: 55 + 144 = 199
144 + 24x + x2 = 199

Factor a perfect square on the left side:
(x + 12)(x + 12) = 199

Calculate the square root of the right side: 14.10673598

Break this problem into two subproblems by setting 
(x + 12) equal to 14.10673598 and -14.10673598.

Subproblem 1

x + 12 = 14.10673598 Simplifying x + 12 = 14.10673598 Reorder the terms: 12 + x = 14.10673598 Solving 12 + x = 14.10673598 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + x = 14.10673598 + -12 Combine like terms: 12 + -12 = 0 0 + x = 14.10673598 + -12 x = 14.10673598 + -12 Combine like terms: 14.10673598 + -12 = 2.10673598 x = 2.10673598 Simplifying x = 2.10673598

Subproblem 2

x + 12 = -14.10673598 Simplifying x + 12 = -14.10673598 Reorder the terms: 12 + x = -14.10673598 Solving 12 + x = -14.10673598 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + x = -14.10673598 + -12 Combine like terms: 12 + -12 = 0 0 + x = -14.10673598 + -12 x = -14.10673598 + -12 Combine like terms: -14.10673598 + -12 = -26.10673598 x = -26.10673598 Simplifying x = -26.10673598

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.10673598, -26.10673598}

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