415=10x+4.9x^2

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Solution for 415=10x+4.9x^2 equation:


Simplifying
415 = 10x + 4.9x2

Solving
415 = 10x + 4.9x2

Solving for variable 'x'.

Reorder the terms:
415 + -10x + -4.9x2 = 10x + -10x + 4.9x2 + -4.9x2

Combine like terms: 10x + -10x = 0
415 + -10x + -4.9x2 = 0 + 4.9x2 + -4.9x2
415 + -10x + -4.9x2 = 4.9x2 + -4.9x2

Combine like terms: 4.9x2 + -4.9x2 = 0.0
415 + -10x + -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'.
-84.69387755 + 2.040816327x + x2 = 0

Move the constant term to the right:

Add '84.69387755' to each side of the equation.
-84.69387755 + 2.040816327x + 84.69387755 + x2 = 0 + 84.69387755

Reorder the terms:
-84.69387755 + 84.69387755 + 2.040816327x + x2 = 0 + 84.69387755

Combine like terms: -84.69387755 + 84.69387755 = 0.00000000
0.00000000 + 2.040816327x + x2 = 0 + 84.69387755
2.040816327x + x2 = 0 + 84.69387755

Combine like terms: 0 + 84.69387755 = 84.69387755
2.040816327x + x2 = 84.69387755

The x term is 2.040816327x.  Take half its coefficient (1.020408164).
Square it (1.041232821) and add it to both sides.

Add '1.041232821' to each side of the equation.
2.040816327x + 1.041232821 + x2 = 84.69387755 + 1.041232821

Reorder the terms:
1.041232821 + 2.040816327x + x2 = 84.69387755 + 1.041232821

Combine like terms: 84.69387755 + 1.041232821 = 85.735110371
1.041232821 + 2.040816327x + x2 = 85.735110371

Factor a perfect square on the left side:
(x + 1.020408164)(x + 1.020408164) = 85.735110371

Calculate the square root of the right side: 9.259325589

Break this problem into two subproblems by setting 
(x + 1.020408164) equal to 9.259325589 and -9.259325589.

Subproblem 1

x + 1.020408164 = 9.259325589 Simplifying x + 1.020408164 = 9.259325589 Reorder the terms: 1.020408164 + x = 9.259325589 Solving 1.020408164 + x = 9.259325589 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.020408164' to each side of the equation. 1.020408164 + -1.020408164 + x = 9.259325589 + -1.020408164 Combine like terms: 1.020408164 + -1.020408164 = 0.000000000 0.000000000 + x = 9.259325589 + -1.020408164 x = 9.259325589 + -1.020408164 Combine like terms: 9.259325589 + -1.020408164 = 8.238917425 x = 8.238917425 Simplifying x = 8.238917425

Subproblem 2

x + 1.020408164 = -9.259325589 Simplifying x + 1.020408164 = -9.259325589 Reorder the terms: 1.020408164 + x = -9.259325589 Solving 1.020408164 + x = -9.259325589 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.020408164' to each side of the equation. 1.020408164 + -1.020408164 + x = -9.259325589 + -1.020408164 Combine like terms: 1.020408164 + -1.020408164 = 0.000000000 0.000000000 + x = -9.259325589 + -1.020408164 x = -9.259325589 + -1.020408164 Combine like terms: -9.259325589 + -1.020408164 = -10.279733753 x = -10.279733753 Simplifying x = -10.279733753

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.238917425, -10.279733753}

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