x^2+0.5x-7.5=20

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Solution for x^2+0.5x-7.5=20 equation:


Simplifying
x2 + 0.5x + -7.5 = 20

Reorder the terms:
-7.5 + 0.5x + x2 = 20

Solving
-7.5 + 0.5x + x2 = 20

Solving for variable 'x'.

Reorder the terms:
-7.5 + -20 + 0.5x + x2 = 20 + -20

Combine like terms: -7.5 + -20 = -27.5
-27.5 + 0.5x + x2 = 20 + -20

Combine like terms: 20 + -20 = 0
-27.5 + 0.5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '27.5' to each side of the equation.
-27.5 + 0.5x + 27.5 + x2 = 0 + 27.5

Reorder the terms:
-27.5 + 27.5 + 0.5x + x2 = 0 + 27.5

Combine like terms: -27.5 + 27.5 = 0.0
0.0 + 0.5x + x2 = 0 + 27.5
0.5x + x2 = 0 + 27.5

Combine like terms: 0 + 27.5 = 27.5
0.5x + x2 = 27.5

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 = 27.5 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 27.5 + 0.0625

Combine like terms: 27.5 + 0.0625 = 27.5625
0.0625 + 0.5x + x2 = 27.5625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 27.5625

Calculate the square root of the right side: 5.25

Break this problem into two subproblems by setting 
(x + 0.25) equal to 5.25 and -5.25.

Subproblem 1

x + 0.25 = 5.25 Simplifying x + 0.25 = 5.25 Reorder the terms: 0.25 + x = 5.25 Solving 0.25 + x = 5.25 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 = 5.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.25 + -0.25 x = 5.25 + -0.25 Combine like terms: 5.25 + -0.25 = 5 x = 5 Simplifying x = 5

Subproblem 2

x + 0.25 = -5.25 Simplifying x + 0.25 = -5.25 Reorder the terms: 0.25 + x = -5.25 Solving 0.25 + x = -5.25 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 = -5.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.25 + -0.25 x = -5.25 + -0.25 Combine like terms: -5.25 + -0.25 = -5.5 x = -5.5 Simplifying x = -5.5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5, -5.5}

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