.5x^2+x=7.5

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Solution for .5x^2+x=7.5 equation:


Simplifying
0.5x2 + x = 7.5

Reorder the terms:
x + 0.5x2 = 7.5

Solving
x + 0.5x2 = 7.5

Solving for variable 'x'.

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

Combine like terms: 7.5 + -7.5 = 0.0
-7.5 + x + 0.5x2 = 0.0

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

Divide each side by '0.5'.
-15 + 2x + x2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 2x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + 2x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 2x + x2 = 0 + 15
2x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
2x + x2 = 15

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 = 15 + 0.25

Reorder the terms:
0.25 + 2x + x2 = 15 + 0.25

Combine like terms: 15 + 0.25 = 15.25
0.25 + 2x + x2 = 15.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 15.25

Calculate the square root of the right side: 3.905124838

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.905124838 and -3.905124838.

Subproblem 1

x + 0.5 = 3.905124838 Simplifying x + 0.5 = 3.905124838 Reorder the terms: 0.5 + x = 3.905124838 Solving 0.5 + x = 3.905124838 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 = 3.905124838 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.905124838 + -0.5 x = 3.905124838 + -0.5 Combine like terms: 3.905124838 + -0.5 = 3.405124838 x = 3.405124838 Simplifying x = 3.405124838

Subproblem 2

x + 0.5 = -3.905124838 Simplifying x + 0.5 = -3.905124838 Reorder the terms: 0.5 + x = -3.905124838 Solving 0.5 + x = -3.905124838 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 = -3.905124838 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.905124838 + -0.5 x = -3.905124838 + -0.5 Combine like terms: -3.905124838 + -0.5 = -4.405124838 x = -4.405124838 Simplifying x = -4.405124838

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.405124838, -4.405124838}

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