0.5x+3=x^2-4

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Solution for 0.5x+3=x^2-4 equation:


Simplifying
0.5x + 3 = x2 + -4

Reorder the terms:
3 + 0.5x = x2 + -4

Reorder the terms:
3 + 0.5x = -4 + x2

Solving
3 + 0.5x = -4 + x2

Solving for variable 'x'.

Reorder the terms:
3 + 4 + 0.5x + -1x2 = -4 + x2 + 4 + -1x2

Combine like terms: 3 + 4 = 7
7 + 0.5x + -1x2 = -4 + x2 + 4 + -1x2

Reorder the terms:
7 + 0.5x + -1x2 = -4 + 4 + x2 + -1x2

Combine like terms: -4 + 4 = 0
7 + 0.5x + -1x2 = 0 + x2 + -1x2
7 + 0.5x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
7 + 0.5x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-7 + -0.5x + x2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -0.5x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + -0.5x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -0.5x + x2 = 0 + 7
-0.5x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
-0.5x + x2 = 7

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 7 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 7 + 0.0625

Combine like terms: 7 + 0.0625 = 7.0625
0.0625 + -0.5x + x2 = 7.0625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(x + -0.25) equal to 2.657536453 and -2.657536453.

Subproblem 1

x + -0.25 = 2.657536453 Simplifying x + -0.25 = 2.657536453 Reorder the terms: -0.25 + x = 2.657536453 Solving -0.25 + x = 2.657536453 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 2.657536453 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 2.657536453 + 0.25 x = 2.657536453 + 0.25 Combine like terms: 2.657536453 + 0.25 = 2.907536453 x = 2.907536453 Simplifying x = 2.907536453

Subproblem 2

x + -0.25 = -2.657536453 Simplifying x + -0.25 = -2.657536453 Reorder the terms: -0.25 + x = -2.657536453 Solving -0.25 + x = -2.657536453 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -2.657536453 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -2.657536453 + 0.25 x = -2.657536453 + 0.25 Combine like terms: -2.657536453 + 0.25 = -2.407536453 x = -2.407536453 Simplifying x = -2.407536453

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.907536453, -2.407536453}

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