x+23*x^2=487

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Solution for x+23*x^2=487 equation:


Simplifying
x + 23x2 = 487

Solving
x + 23x2 = 487

Solving for variable 'x'.

Reorder the terms:
-487 + x + 23x2 = 487 + -487

Combine like terms: 487 + -487 = 0
-487 + x + 23x2 = 0

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

Divide each side by '23'.
-21.17391304 + 0.04347826087x + x2 = 0

Move the constant term to the right:

Add '21.17391304' to each side of the equation.
-21.17391304 + 0.04347826087x + 21.17391304 + x2 = 0 + 21.17391304

Reorder the terms:
-21.17391304 + 21.17391304 + 0.04347826087x + x2 = 0 + 21.17391304

Combine like terms: -21.17391304 + 21.17391304 = 0.00000000
0.00000000 + 0.04347826087x + x2 = 0 + 21.17391304
0.04347826087x + x2 = 0 + 21.17391304

Combine like terms: 0 + 21.17391304 = 21.17391304
0.04347826087x + x2 = 21.17391304

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.04347826087x + 0.25 + x2 = 21.17391304 + 0.25

Reorder the terms:
0.25 + 0.04347826087x + x2 = 21.17391304 + 0.25

Combine like terms: 21.17391304 + 0.25 = 21.42391304
0.25 + 0.04347826087x + x2 = 21.42391304

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 21.42391304

Calculate the square root of the right side: 4.628597308

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.628597308 and -4.628597308.

Subproblem 1

x + 0.5 = 4.628597308 Simplifying x + 0.5 = 4.628597308 Reorder the terms: 0.5 + x = 4.628597308 Solving 0.5 + x = 4.628597308 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 4.628597308 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 4.628597308 + -0.5 x = 4.628597308 + -0.5 Combine like terms: 4.628597308 + -0.5 = 4.128597308 x = 4.128597308 Simplifying x = 4.128597308

Subproblem 2

x + 0.5 = -4.628597308 Simplifying x + 0.5 = -4.628597308 Reorder the terms: 0.5 + x = -4.628597308 Solving 0.5 + x = -4.628597308 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -4.628597308 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -4.628597308 + -0.5 x = -4.628597308 + -0.5 Combine like terms: -4.628597308 + -0.5 = -5.128597308 x = -5.128597308 Simplifying x = -5.128597308

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.128597308, -5.128597308}

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