-.5x^2+x+24=0

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Solution for -.5x^2+x+24=0 equation:


Simplifying
-0.5x2 + x + 24 = 0

Reorder the terms:
24 + x + -0.5x2 = 0

Solving
24 + x + -0.5x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-0.5 the coefficient of the squared term: 

Divide each side by '-0.5'.
-48 + -2x + x2 = 0

Move the constant term to the right:

Add '48' to each side of the equation.
-48 + -2x + 48 + x2 = 0 + 48

Reorder the terms:
-48 + 48 + -2x + x2 = 0 + 48

Combine like terms: -48 + 48 = 0
0 + -2x + x2 = 0 + 48
-2x + x2 = 0 + 48

Combine like terms: 0 + 48 = 48
-2x + x2 = 48

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.
-2x + 0.25 + x2 = 48 + 0.25

Reorder the terms:
0.25 + -2x + x2 = 48 + 0.25

Combine like terms: 48 + 0.25 = 48.25
0.25 + -2x + x2 = 48.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 48.25

Calculate the square root of the right side: 6.946221995

Break this problem into two subproblems by setting 
(x + 0.5) equal to 6.946221995 and -6.946221995.

Subproblem 1

x + 0.5 = 6.946221995 Simplifying x + 0.5 = 6.946221995 Reorder the terms: 0.5 + x = 6.946221995 Solving 0.5 + x = 6.946221995 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 = 6.946221995 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.946221995 + -0.5 x = 6.946221995 + -0.5 Combine like terms: 6.946221995 + -0.5 = 6.446221995 x = 6.446221995 Simplifying x = 6.446221995

Subproblem 2

x + 0.5 = -6.946221995 Simplifying x + 0.5 = -6.946221995 Reorder the terms: 0.5 + x = -6.946221995 Solving 0.5 + x = -6.946221995 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 = -6.946221995 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.946221995 + -0.5 x = -6.946221995 + -0.5 Combine like terms: -6.946221995 + -0.5 = -7.446221995 x = -7.446221995 Simplifying x = -7.446221995

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.446221995, -7.446221995}

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