0=x^2+2.25x+0.75

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Solution for 0=x^2+2.25x+0.75 equation:


Simplifying
0 = x2 + 2.25x + 0.75

Reorder the terms:
0 = 0.75 + 2.25x + x2

Solving
0 = 0.75 + 2.25x + x2

Solving for variable 'x'.

Combine like terms: 0 + -0.75 = -0.75
-0.75 + -2.25x + -1x2 = 0.75 + 2.25x + x2 + -0.75 + -2.25x + -1x2

Reorder the terms:
-0.75 + -2.25x + -1x2 = 0.75 + -0.75 + 2.25x + -2.25x + x2 + -1x2

Combine like terms: 0.75 + -0.75 = 0.00
-0.75 + -2.25x + -1x2 = 0.00 + 2.25x + -2.25x + x2 + -1x2
-0.75 + -2.25x + -1x2 = 2.25x + -2.25x + x2 + -1x2

Combine like terms: 2.25x + -2.25x = 0.00
-0.75 + -2.25x + -1x2 = 0.00 + x2 + -1x2
-0.75 + -2.25x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-0.75 + -2.25x + -1x2 = 0

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

Divide each side by '-1'.
0.75 + 2.25x + x2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + 2.25x + -0.75 + x2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + 2.25x + x2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + 2.25x + x2 = 0 + -0.75
2.25x + x2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
2.25x + x2 = -0.75

The x term is 2.25x.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25x + 1.265625 + x2 = -0.75 + 1.265625

Reorder the terms:
1.265625 + 2.25x + x2 = -0.75 + 1.265625

Combine like terms: -0.75 + 1.265625 = 0.515625
1.265625 + 2.25x + x2 = 0.515625

Factor a perfect square on the left side:
(x + 1.125)(x + 1.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(x + 1.125) equal to 0.718070331 and -0.718070331.

Subproblem 1

x + 1.125 = 0.718070331 Simplifying x + 1.125 = 0.718070331 Reorder the terms: 1.125 + x = 0.718070331 Solving 1.125 + x = 0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = 0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = 0.718070331 + -1.125 x = 0.718070331 + -1.125 Combine like terms: 0.718070331 + -1.125 = -0.406929669 x = -0.406929669 Simplifying x = -0.406929669

Subproblem 2

x + 1.125 = -0.718070331 Simplifying x + 1.125 = -0.718070331 Reorder the terms: 1.125 + x = -0.718070331 Solving 1.125 + x = -0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = -0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = -0.718070331 + -1.125 x = -0.718070331 + -1.125 Combine like terms: -0.718070331 + -1.125 = -1.843070331 x = -1.843070331 Simplifying x = -1.843070331

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.406929669, -1.843070331}

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