19.5*x^2-420*x+1600=0

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Solution for 19.5*x^2-420*x+1600=0 equation:


Simplifying
19.5x2 + -420x + 1600 = 0

Reorder the terms:
1600 + -420x + 19.5x2 = 0

Solving
1600 + -420x + 19.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '19.5'.
82.05128205 + -21.53846154x + x2 = 0

Move the constant term to the right:

Add '-82.05128205' to each side of the equation.
82.05128205 + -21.53846154x + -82.05128205 + x2 = 0 + -82.05128205

Reorder the terms:
82.05128205 + -82.05128205 + -21.53846154x + x2 = 0 + -82.05128205

Combine like terms: 82.05128205 + -82.05128205 = 0.00000000
0.00000000 + -21.53846154x + x2 = 0 + -82.05128205
-21.53846154x + x2 = 0 + -82.05128205

Combine like terms: 0 + -82.05128205 = -82.05128205
-21.53846154x + x2 = -82.05128205

The x term is -21.53846154x.  Take half its coefficient (-10.76923077).
Square it (115.9763314) and add it to both sides.

Add '115.9763314' to each side of the equation.
-21.53846154x + 115.9763314 + x2 = -82.05128205 + 115.9763314

Reorder the terms:
115.9763314 + -21.53846154x + x2 = -82.05128205 + 115.9763314

Combine like terms: -82.05128205 + 115.9763314 = 33.92504935
115.9763314 + -21.53846154x + x2 = 33.92504935

Factor a perfect square on the left side:
(x + -10.76923077)(x + -10.76923077) = 33.92504935

Calculate the square root of the right side: 5.824521384

Break this problem into two subproblems by setting 
(x + -10.76923077) equal to 5.824521384 and -5.824521384.

Subproblem 1

x + -10.76923077 = 5.824521384 Simplifying x + -10.76923077 = 5.824521384 Reorder the terms: -10.76923077 + x = 5.824521384 Solving -10.76923077 + x = 5.824521384 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.76923077' to each side of the equation. -10.76923077 + 10.76923077 + x = 5.824521384 + 10.76923077 Combine like terms: -10.76923077 + 10.76923077 = 0.00000000 0.00000000 + x = 5.824521384 + 10.76923077 x = 5.824521384 + 10.76923077 Combine like terms: 5.824521384 + 10.76923077 = 16.593752154 x = 16.593752154 Simplifying x = 16.593752154

Subproblem 2

x + -10.76923077 = -5.824521384 Simplifying x + -10.76923077 = -5.824521384 Reorder the terms: -10.76923077 + x = -5.824521384 Solving -10.76923077 + x = -5.824521384 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.76923077' to each side of the equation. -10.76923077 + 10.76923077 + x = -5.824521384 + 10.76923077 Combine like terms: -10.76923077 + 10.76923077 = 0.00000000 0.00000000 + x = -5.824521384 + 10.76923077 x = -5.824521384 + 10.76923077 Combine like terms: -5.824521384 + 10.76923077 = 4.944709386 x = 4.944709386 Simplifying x = 4.944709386

Solution

The solution to the problem is based on the solutions from the subproblems. x = {16.593752154, 4.944709386}

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