4x^2-40x+97=0

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Solution for 4x^2-40x+97=0 equation:


Simplifying
4x2 + -40x + 97 = 0

Reorder the terms:
97 + -40x + 4x2 = 0

Solving
97 + -40x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
24.25 + -10x + x2 = 0

Move the constant term to the right:

Add '-24.25' to each side of the equation.
24.25 + -10x + -24.25 + x2 = 0 + -24.25

Reorder the terms:
24.25 + -24.25 + -10x + x2 = 0 + -24.25

Combine like terms: 24.25 + -24.25 = 0.00
0.00 + -10x + x2 = 0 + -24.25
-10x + x2 = 0 + -24.25

Combine like terms: 0 + -24.25 = -24.25
-10x + x2 = -24.25

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = -24.25 + 25

Reorder the terms:
25 + -10x + x2 = -24.25 + 25

Combine like terms: -24.25 + 25 = 0.75
25 + -10x + x2 = 0.75

Factor a perfect square on the left side:
(x + -5)(x + -5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x + -5) equal to 0.866025404 and -0.866025404.

Subproblem 1

x + -5 = 0.866025404 Simplifying x + -5 = 0.866025404 Reorder the terms: -5 + x = 0.866025404 Solving -5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 0.866025404 + 5 Combine like terms: -5 + 5 = 0 0 + x = 0.866025404 + 5 x = 0.866025404 + 5 Combine like terms: 0.866025404 + 5 = 5.866025404 x = 5.866025404 Simplifying x = 5.866025404

Subproblem 2

x + -5 = -0.866025404 Simplifying x + -5 = -0.866025404 Reorder the terms: -5 + x = -0.866025404 Solving -5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -0.866025404 + 5 Combine like terms: -5 + 5 = 0 0 + x = -0.866025404 + 5 x = -0.866025404 + 5 Combine like terms: -0.866025404 + 5 = 4.133974596 x = 4.133974596 Simplifying x = 4.133974596

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.866025404, 4.133974596}

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