0=-4.9x^2+13x+500

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Solution for 0=-4.9x^2+13x+500 equation:


Simplifying
0 = -4.9x2 + 13x + 500

Reorder the terms:
0 = 500 + 13x + -4.9x2

Solving
0 = 500 + 13x + -4.9x2

Solving for variable 'x'.

Combine like terms: 0 + -500 = -500
-500 + -13x + 4.9x2 = 500 + 13x + -4.9x2 + -500 + -13x + 4.9x2

Reorder the terms:
-500 + -13x + 4.9x2 = 500 + -500 + 13x + -13x + -4.9x2 + 4.9x2

Combine like terms: 500 + -500 = 0
-500 + -13x + 4.9x2 = 0 + 13x + -13x + -4.9x2 + 4.9x2
-500 + -13x + 4.9x2 = 13x + -13x + -4.9x2 + 4.9x2

Combine like terms: 13x + -13x = 0
-500 + -13x + 4.9x2 = 0 + -4.9x2 + 4.9x2
-500 + -13x + 4.9x2 = -4.9x2 + 4.9x2

Combine like terms: -4.9x2 + 4.9x2 = 0.0
-500 + -13x + 4.9x2 = 0.0

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

Divide each side by '4.9'.
-102.0408163 + -2.653061224x + x2 = 0

Move the constant term to the right:

Add '102.0408163' to each side of the equation.
-102.0408163 + -2.653061224x + 102.0408163 + x2 = 0 + 102.0408163

Reorder the terms:
-102.0408163 + 102.0408163 + -2.653061224x + x2 = 0 + 102.0408163

Combine like terms: -102.0408163 + 102.0408163 = 0.0000000
0.0000000 + -2.653061224x + x2 = 0 + 102.0408163
-2.653061224x + x2 = 0 + 102.0408163

Combine like terms: 0 + 102.0408163 = 102.0408163
-2.653061224x + x2 = 102.0408163

The x term is -2.653061224x.  Take half its coefficient (-1.326530612).
Square it (1.759683465) and add it to both sides.

Add '1.759683465' to each side of the equation.
-2.653061224x + 1.759683465 + x2 = 102.0408163 + 1.759683465

Reorder the terms:
1.759683465 + -2.653061224x + x2 = 102.0408163 + 1.759683465

Combine like terms: 102.0408163 + 1.759683465 = 103.800499765
1.759683465 + -2.653061224x + x2 = 103.800499765

Factor a perfect square on the left side:
(x + -1.326530612)(x + -1.326530612) = 103.800499765

Calculate the square root of the right side: 10.188253028

Break this problem into two subproblems by setting 
(x + -1.326530612) equal to 10.188253028 and -10.188253028.

Subproblem 1

x + -1.326530612 = 10.188253028 Simplifying x + -1.326530612 = 10.188253028 Reorder the terms: -1.326530612 + x = 10.188253028 Solving -1.326530612 + x = 10.188253028 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.326530612' to each side of the equation. -1.326530612 + 1.326530612 + x = 10.188253028 + 1.326530612 Combine like terms: -1.326530612 + 1.326530612 = 0.000000000 0.000000000 + x = 10.188253028 + 1.326530612 x = 10.188253028 + 1.326530612 Combine like terms: 10.188253028 + 1.326530612 = 11.51478364 x = 11.51478364 Simplifying x = 11.51478364

Subproblem 2

x + -1.326530612 = -10.188253028 Simplifying x + -1.326530612 = -10.188253028 Reorder the terms: -1.326530612 + x = -10.188253028 Solving -1.326530612 + x = -10.188253028 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.326530612' to each side of the equation. -1.326530612 + 1.326530612 + x = -10.188253028 + 1.326530612 Combine like terms: -1.326530612 + 1.326530612 = 0.000000000 0.000000000 + x = -10.188253028 + 1.326530612 x = -10.188253028 + 1.326530612 Combine like terms: -10.188253028 + 1.326530612 = -8.861722416 x = -8.861722416 Simplifying x = -8.861722416

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11.51478364, -8.861722416}

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