61x^2+20x+4=90

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Solution for 61x^2+20x+4=90 equation:


Simplifying
61x2 + 20x + 4 = 90

Reorder the terms:
4 + 20x + 61x2 = 90

Solving
4 + 20x + 61x2 = 90

Solving for variable 'x'.

Reorder the terms:
4 + -90 + 20x + 61x2 = 90 + -90

Combine like terms: 4 + -90 = -86
-86 + 20x + 61x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-86 + 20x + 61x2 = 0

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

Divide each side by '61'.
-1.409836066 + 0.3278688525x + x2 = 0

Move the constant term to the right:

Add '1.409836066' to each side of the equation.
-1.409836066 + 0.3278688525x + 1.409836066 + x2 = 0 + 1.409836066

Reorder the terms:
-1.409836066 + 1.409836066 + 0.3278688525x + x2 = 0 + 1.409836066

Combine like terms: -1.409836066 + 1.409836066 = 0.000000000
0.000000000 + 0.3278688525x + x2 = 0 + 1.409836066
0.3278688525x + x2 = 0 + 1.409836066

Combine like terms: 0 + 1.409836066 = 1.409836066
0.3278688525x + x2 = 1.409836066

The x term is 0.3278688525x.  Take half its coefficient (0.1639344263).
Square it (0.02687449613) and add it to both sides.

Add '0.02687449613' to each side of the equation.
0.3278688525x + 0.02687449613 + x2 = 1.409836066 + 0.02687449613

Reorder the terms:
0.02687449613 + 0.3278688525x + x2 = 1.409836066 + 0.02687449613

Combine like terms: 1.409836066 + 0.02687449613 = 1.43671056213
0.02687449613 + 0.3278688525x + x2 = 1.43671056213

Factor a perfect square on the left side:
(x + 0.1639344263)(x + 0.1639344263) = 1.43671056213

Calculate the square root of the right side: 1.198628617

Break this problem into two subproblems by setting 
(x + 0.1639344263) equal to 1.198628617 and -1.198628617.

Subproblem 1

x + 0.1639344263 = 1.198628617 Simplifying x + 0.1639344263 = 1.198628617 Reorder the terms: 0.1639344263 + x = 1.198628617 Solving 0.1639344263 + x = 1.198628617 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1639344263' to each side of the equation. 0.1639344263 + -0.1639344263 + x = 1.198628617 + -0.1639344263 Combine like terms: 0.1639344263 + -0.1639344263 = 0.0000000000 0.0000000000 + x = 1.198628617 + -0.1639344263 x = 1.198628617 + -0.1639344263 Combine like terms: 1.198628617 + -0.1639344263 = 1.0346941907 x = 1.0346941907 Simplifying x = 1.0346941907

Subproblem 2

x + 0.1639344263 = -1.198628617 Simplifying x + 0.1639344263 = -1.198628617 Reorder the terms: 0.1639344263 + x = -1.198628617 Solving 0.1639344263 + x = -1.198628617 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1639344263' to each side of the equation. 0.1639344263 + -0.1639344263 + x = -1.198628617 + -0.1639344263 Combine like terms: 0.1639344263 + -0.1639344263 = 0.0000000000 0.0000000000 + x = -1.198628617 + -0.1639344263 x = -1.198628617 + -0.1639344263 Combine like terms: -1.198628617 + -0.1639344263 = -1.3625630433 x = -1.3625630433 Simplifying x = -1.3625630433

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.0346941907, -1.3625630433}

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