0=-4.9x^2-5x+450

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Solution for 0=-4.9x^2-5x+450 equation:


Simplifying
0 = -4.9x2 + -5x + 450

Reorder the terms:
0 = 450 + -5x + -4.9x2

Solving
0 = 450 + -5x + -4.9x2

Solving for variable 'x'.

Combine like terms: 0 + -450 = -450
-450 + 5x + 4.9x2 = 450 + -5x + -4.9x2 + -450 + 5x + 4.9x2

Reorder the terms:
-450 + 5x + 4.9x2 = 450 + -450 + -5x + 5x + -4.9x2 + 4.9x2

Combine like terms: 450 + -450 = 0
-450 + 5x + 4.9x2 = 0 + -5x + 5x + -4.9x2 + 4.9x2
-450 + 5x + 4.9x2 = -5x + 5x + -4.9x2 + 4.9x2

Combine like terms: -5x + 5x = 0
-450 + 5x + 4.9x2 = 0 + -4.9x2 + 4.9x2
-450 + 5x + 4.9x2 = -4.9x2 + 4.9x2

Combine like terms: -4.9x2 + 4.9x2 = 0.0
-450 + 5x + 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'.
-91.83673469 + 1.020408163x + x2 = 0

Move the constant term to the right:

Add '91.83673469' to each side of the equation.
-91.83673469 + 1.020408163x + 91.83673469 + x2 = 0 + 91.83673469

Reorder the terms:
-91.83673469 + 91.83673469 + 1.020408163x + x2 = 0 + 91.83673469

Combine like terms: -91.83673469 + 91.83673469 = 0.00000000
0.00000000 + 1.020408163x + x2 = 0 + 91.83673469
1.020408163x + x2 = 0 + 91.83673469

Combine like terms: 0 + 91.83673469 = 91.83673469
1.020408163x + x2 = 91.83673469

The x term is 1.020408163x.  Take half its coefficient (0.5102040815).
Square it (0.2603082048) and add it to both sides.

Add '0.2603082048' to each side of the equation.
1.020408163x + 0.2603082048 + x2 = 91.83673469 + 0.2603082048

Reorder the terms:
0.2603082048 + 1.020408163x + x2 = 91.83673469 + 0.2603082048

Combine like terms: 91.83673469 + 0.2603082048 = 92.0970428948
0.2603082048 + 1.020408163x + x2 = 92.0970428948

Factor a perfect square on the left side:
(x + 0.5102040815)(x + 0.5102040815) = 92.0970428948

Calculate the square root of the right side: 9.596720424

Break this problem into two subproblems by setting 
(x + 0.5102040815) equal to 9.596720424 and -9.596720424.

Subproblem 1

x + 0.5102040815 = 9.596720424 Simplifying x + 0.5102040815 = 9.596720424 Reorder the terms: 0.5102040815 + x = 9.596720424 Solving 0.5102040815 + x = 9.596720424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5102040815' to each side of the equation. 0.5102040815 + -0.5102040815 + x = 9.596720424 + -0.5102040815 Combine like terms: 0.5102040815 + -0.5102040815 = 0.0000000000 0.0000000000 + x = 9.596720424 + -0.5102040815 x = 9.596720424 + -0.5102040815 Combine like terms: 9.596720424 + -0.5102040815 = 9.0865163425 x = 9.0865163425 Simplifying x = 9.0865163425

Subproblem 2

x + 0.5102040815 = -9.596720424 Simplifying x + 0.5102040815 = -9.596720424 Reorder the terms: 0.5102040815 + x = -9.596720424 Solving 0.5102040815 + x = -9.596720424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5102040815' to each side of the equation. 0.5102040815 + -0.5102040815 + x = -9.596720424 + -0.5102040815 Combine like terms: 0.5102040815 + -0.5102040815 = 0.0000000000 0.0000000000 + x = -9.596720424 + -0.5102040815 x = -9.596720424 + -0.5102040815 Combine like terms: -9.596720424 + -0.5102040815 = -10.1069245055 x = -10.1069245055 Simplifying x = -10.1069245055

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.0865163425, -10.1069245055}

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