224x-x^2=784

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Solution for 224x-x^2=784 equation:


Simplifying
224x + -1x2 = 784

Solving
224x + -1x2 = 784

Solving for variable 'x'.

Reorder the terms:
-784 + 224x + -1x2 = 784 + -784

Combine like terms: 784 + -784 = 0
-784 + 224x + -1x2 = 0

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

Divide each side by '-1'.
784 + -224x + x2 = 0

Move the constant term to the right:

Add '-784' to each side of the equation.
784 + -224x + -784 + x2 = 0 + -784

Reorder the terms:
784 + -784 + -224x + x2 = 0 + -784

Combine like terms: 784 + -784 = 0
0 + -224x + x2 = 0 + -784
-224x + x2 = 0 + -784

Combine like terms: 0 + -784 = -784
-224x + x2 = -784

The x term is -224x.  Take half its coefficient (-112).
Square it (12544) and add it to both sides.

Add '12544' to each side of the equation.
-224x + 12544 + x2 = -784 + 12544

Reorder the terms:
12544 + -224x + x2 = -784 + 12544

Combine like terms: -784 + 12544 = 11760
12544 + -224x + x2 = 11760

Factor a perfect square on the left side:
(x + -112)(x + -112) = 11760

Calculate the square root of the right side: 108.443533694

Break this problem into two subproblems by setting 
(x + -112) equal to 108.443533694 and -108.443533694.

Subproblem 1

x + -112 = 108.443533694 Simplifying x + -112 = 108.443533694 Reorder the terms: -112 + x = 108.443533694 Solving -112 + x = 108.443533694 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '112' to each side of the equation. -112 + 112 + x = 108.443533694 + 112 Combine like terms: -112 + 112 = 0 0 + x = 108.443533694 + 112 x = 108.443533694 + 112 Combine like terms: 108.443533694 + 112 = 220.443533694 x = 220.443533694 Simplifying x = 220.443533694

Subproblem 2

x + -112 = -108.443533694 Simplifying x + -112 = -108.443533694 Reorder the terms: -112 + x = -108.443533694 Solving -112 + x = -108.443533694 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '112' to each side of the equation. -112 + 112 + x = -108.443533694 + 112 Combine like terms: -112 + 112 = 0 0 + x = -108.443533694 + 112 x = -108.443533694 + 112 Combine like terms: -108.443533694 + 112 = 3.556466306 x = 3.556466306 Simplifying x = 3.556466306

Solution

The solution to the problem is based on the solutions from the subproblems. x = {220.443533694, 3.556466306}

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