500=1.36(x*x)+27.8x+304

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Solution for 500=1.36(x*x)+27.8x+304 equation:


Simplifying
500 = 1.36(x * x) + 27.8x + 304

Multiply x * x
500 = 1.36(x2) + 27.8x + 304

Reorder the terms:
500 = 304 + 27.8x + 1.36x2

Solving
500 = 304 + 27.8x + 1.36x2

Solving for variable 'x'.

Combine like terms: 500 + -304 = 196
196 + -27.8x + -1.36x2 = 304 + 27.8x + 1.36x2 + -304 + -27.8x + -1.36x2

Reorder the terms:
196 + -27.8x + -1.36x2 = 304 + -304 + 27.8x + -27.8x + 1.36x2 + -1.36x2

Combine like terms: 304 + -304 = 0
196 + -27.8x + -1.36x2 = 0 + 27.8x + -27.8x + 1.36x2 + -1.36x2
196 + -27.8x + -1.36x2 = 27.8x + -27.8x + 1.36x2 + -1.36x2

Combine like terms: 27.8x + -27.8x = 0.0
196 + -27.8x + -1.36x2 = 0.0 + 1.36x2 + -1.36x2
196 + -27.8x + -1.36x2 = 1.36x2 + -1.36x2

Combine like terms: 1.36x2 + -1.36x2 = 0.00
196 + -27.8x + -1.36x2 = 0.00

Begin completing the square.  Divide all terms by
-1.36 the coefficient of the squared term: 

Divide each side by '-1.36'.
-144.1176471 + 20.44117647x + x2 = 0

Move the constant term to the right:

Add '144.1176471' to each side of the equation.
-144.1176471 + 20.44117647x + 144.1176471 + x2 = 0 + 144.1176471

Reorder the terms:
-144.1176471 + 144.1176471 + 20.44117647x + x2 = 0 + 144.1176471

Combine like terms: -144.1176471 + 144.1176471 = 0.0000000
0.0000000 + 20.44117647x + x2 = 0 + 144.1176471
20.44117647x + x2 = 0 + 144.1176471

Combine like terms: 0 + 144.1176471 = 144.1176471
20.44117647x + x2 = 144.1176471

The x term is 20.44117647x.  Take half its coefficient (10.22058824).
Square it (104.4604240) and add it to both sides.

Add '104.4604240' to each side of the equation.
20.44117647x + 104.4604240 + x2 = 144.1176471 + 104.4604240

Reorder the terms:
104.4604240 + 20.44117647x + x2 = 144.1176471 + 104.4604240

Combine like terms: 144.1176471 + 104.4604240 = 248.5780711
104.4604240 + 20.44117647x + x2 = 248.5780711

Factor a perfect square on the left side:
(x + 10.22058824)(x + 10.22058824) = 248.5780711

Calculate the square root of the right side: 15.766358841

Break this problem into two subproblems by setting 
(x + 10.22058824) equal to 15.766358841 and -15.766358841.

Subproblem 1

x + 10.22058824 = 15.766358841 Simplifying x + 10.22058824 = 15.766358841 Reorder the terms: 10.22058824 + x = 15.766358841 Solving 10.22058824 + x = 15.766358841 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.22058824' to each side of the equation. 10.22058824 + -10.22058824 + x = 15.766358841 + -10.22058824 Combine like terms: 10.22058824 + -10.22058824 = 0.00000000 0.00000000 + x = 15.766358841 + -10.22058824 x = 15.766358841 + -10.22058824 Combine like terms: 15.766358841 + -10.22058824 = 5.545770601 x = 5.545770601 Simplifying x = 5.545770601

Subproblem 2

x + 10.22058824 = -15.766358841 Simplifying x + 10.22058824 = -15.766358841 Reorder the terms: 10.22058824 + x = -15.766358841 Solving 10.22058824 + x = -15.766358841 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.22058824' to each side of the equation. 10.22058824 + -10.22058824 + x = -15.766358841 + -10.22058824 Combine like terms: 10.22058824 + -10.22058824 = 0.00000000 0.00000000 + x = -15.766358841 + -10.22058824 x = -15.766358841 + -10.22058824 Combine like terms: -15.766358841 + -10.22058824 = -25.986947081 x = -25.986947081 Simplifying x = -25.986947081

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.545770601, -25.986947081}

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