3x^2-5x+4=61

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Solution for 3x^2-5x+4=61 equation:


Simplifying
3x2 + -5x + 4 = 61

Reorder the terms:
4 + -5x + 3x2 = 61

Solving
4 + -5x + 3x2 = 61

Solving for variable 'x'.

Reorder the terms:
4 + -61 + -5x + 3x2 = 61 + -61

Combine like terms: 4 + -61 = -57
-57 + -5x + 3x2 = 61 + -61

Combine like terms: 61 + -61 = 0
-57 + -5x + 3x2 = 0

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

Divide each side by '3'.
-19 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + -1.666666667x + 19 + x2 = 0 + 19

Reorder the terms:
-19 + 19 + -1.666666667x + x2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + -1.666666667x + x2 = 0 + 19
-1.666666667x + x2 = 0 + 19

Combine like terms: 0 + 19 = 19
-1.666666667x + x2 = 19

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 19 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 19 + 0.6944444447

Combine like terms: 19 + 0.6944444447 = 19.6944444447
0.6944444447 + -1.666666667x + x2 = 19.6944444447

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 19.6944444447

Calculate the square root of the right side: 4.437842319

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 4.437842319 and -4.437842319.

Subproblem 1

x + -0.8333333335 = 4.437842319 Simplifying x + -0.8333333335 = 4.437842319 Reorder the terms: -0.8333333335 + x = 4.437842319 Solving -0.8333333335 + x = 4.437842319 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 4.437842319 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 4.437842319 + 0.8333333335 x = 4.437842319 + 0.8333333335 Combine like terms: 4.437842319 + 0.8333333335 = 5.2711756525 x = 5.2711756525 Simplifying x = 5.2711756525

Subproblem 2

x + -0.8333333335 = -4.437842319 Simplifying x + -0.8333333335 = -4.437842319 Reorder the terms: -0.8333333335 + x = -4.437842319 Solving -0.8333333335 + x = -4.437842319 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -4.437842319 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -4.437842319 + 0.8333333335 x = -4.437842319 + 0.8333333335 Combine like terms: -4.437842319 + 0.8333333335 = -3.6045089855 x = -3.6045089855 Simplifying x = -3.6045089855

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.2711756525, -3.6045089855}

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