1.682x-.48x^2=.5

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Solution for 1.682x-.48x^2=.5 equation:


Simplifying
1.682x + -0.48x2 = 0.5

Solving
1.682x + -0.48x2 = 0.5

Solving for variable 'x'.

Reorder the terms:
-0.5 + 1.682x + -0.48x2 = 0.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
-0.5 + 1.682x + -0.48x2 = 0.0

Begin completing the square.  Divide all terms by
-0.48 the coefficient of the squared term: 

Divide each side by '-0.48'.
1.041666667 + -3.504166667x + x2 = 0

Move the constant term to the right:

Add '-1.041666667' to each side of the equation.
1.041666667 + -3.504166667x + -1.041666667 + x2 = 0 + -1.041666667

Reorder the terms:
1.041666667 + -1.041666667 + -3.504166667x + x2 = 0 + -1.041666667

Combine like terms: 1.041666667 + -1.041666667 = 0.000000000
0.000000000 + -3.504166667x + x2 = 0 + -1.041666667
-3.504166667x + x2 = 0 + -1.041666667

Combine like terms: 0 + -1.041666667 = -1.041666667
-3.504166667x + x2 = -1.041666667

The x term is -3.504166667x.  Take half its coefficient (-1.752083334).
Square it (3.069796009) and add it to both sides.

Add '3.069796009' to each side of the equation.
-3.504166667x + 3.069796009 + x2 = -1.041666667 + 3.069796009

Reorder the terms:
3.069796009 + -3.504166667x + x2 = -1.041666667 + 3.069796009

Combine like terms: -1.041666667 + 3.069796009 = 2.028129342
3.069796009 + -3.504166667x + x2 = 2.028129342

Factor a perfect square on the left side:
(x + -1.752083334)(x + -1.752083334) = 2.028129342

Calculate the square root of the right side: 1.424124061

Break this problem into two subproblems by setting 
(x + -1.752083334) equal to 1.424124061 and -1.424124061.

Subproblem 1

x + -1.752083334 = 1.424124061 Simplifying x + -1.752083334 = 1.424124061 Reorder the terms: -1.752083334 + x = 1.424124061 Solving -1.752083334 + x = 1.424124061 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.752083334' to each side of the equation. -1.752083334 + 1.752083334 + x = 1.424124061 + 1.752083334 Combine like terms: -1.752083334 + 1.752083334 = 0.000000000 0.000000000 + x = 1.424124061 + 1.752083334 x = 1.424124061 + 1.752083334 Combine like terms: 1.424124061 + 1.752083334 = 3.176207395 x = 3.176207395 Simplifying x = 3.176207395

Subproblem 2

x + -1.752083334 = -1.424124061 Simplifying x + -1.752083334 = -1.424124061 Reorder the terms: -1.752083334 + x = -1.424124061 Solving -1.752083334 + x = -1.424124061 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.752083334' to each side of the equation. -1.752083334 + 1.752083334 + x = -1.424124061 + 1.752083334 Combine like terms: -1.752083334 + 1.752083334 = 0.000000000 0.000000000 + x = -1.424124061 + 1.752083334 x = -1.424124061 + 1.752083334 Combine like terms: -1.424124061 + 1.752083334 = 0.327959273 x = 0.327959273 Simplifying x = 0.327959273

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.176207395, 0.327959273}

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