120=4x^2+5x+1

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Solution for 120=4x^2+5x+1 equation:


Simplifying
120 = 4x2 + 5x + 1

Reorder the terms:
120 = 1 + 5x + 4x2

Solving
120 = 1 + 5x + 4x2

Solving for variable 'x'.

Combine like terms: 120 + -1 = 119
119 + -5x + -4x2 = 1 + 5x + 4x2 + -1 + -5x + -4x2

Reorder the terms:
119 + -5x + -4x2 = 1 + -1 + 5x + -5x + 4x2 + -4x2

Combine like terms: 1 + -1 = 0
119 + -5x + -4x2 = 0 + 5x + -5x + 4x2 + -4x2
119 + -5x + -4x2 = 5x + -5x + 4x2 + -4x2

Combine like terms: 5x + -5x = 0
119 + -5x + -4x2 = 0 + 4x2 + -4x2
119 + -5x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
119 + -5x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-29.75 + 1.25x + x2 = 0

Move the constant term to the right:

Add '29.75' to each side of the equation.
-29.75 + 1.25x + 29.75 + x2 = 0 + 29.75

Reorder the terms:
-29.75 + 29.75 + 1.25x + x2 = 0 + 29.75

Combine like terms: -29.75 + 29.75 = 0.00
0.00 + 1.25x + x2 = 0 + 29.75
1.25x + x2 = 0 + 29.75

Combine like terms: 0 + 29.75 = 29.75
1.25x + x2 = 29.75

The x term is 1.25x.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25x + 0.390625 + x2 = 29.75 + 0.390625

Reorder the terms:
0.390625 + 1.25x + x2 = 29.75 + 0.390625

Combine like terms: 29.75 + 0.390625 = 30.140625
0.390625 + 1.25x + x2 = 30.140625

Factor a perfect square on the left side:
(x + 0.625)(x + 0.625) = 30.140625

Calculate the square root of the right side: 5.490047814

Break this problem into two subproblems by setting 
(x + 0.625) equal to 5.490047814 and -5.490047814.

Subproblem 1

x + 0.625 = 5.490047814 Simplifying x + 0.625 = 5.490047814 Reorder the terms: 0.625 + x = 5.490047814 Solving 0.625 + x = 5.490047814 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 5.490047814 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 5.490047814 + -0.625 x = 5.490047814 + -0.625 Combine like terms: 5.490047814 + -0.625 = 4.865047814 x = 4.865047814 Simplifying x = 4.865047814

Subproblem 2

x + 0.625 = -5.490047814 Simplifying x + 0.625 = -5.490047814 Reorder the terms: 0.625 + x = -5.490047814 Solving 0.625 + x = -5.490047814 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -5.490047814 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -5.490047814 + -0.625 x = -5.490047814 + -0.625 Combine like terms: -5.490047814 + -0.625 = -6.115047814 x = -6.115047814 Simplifying x = -6.115047814

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.865047814, -6.115047814}

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