4x^2-31x+45=0

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Solution for 4x^2-31x+45=0 equation:


Simplifying
4x2 + -31x + 45 = 0

Reorder the terms:
45 + -31x + 4x2 = 0

Solving
45 + -31x + 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'.
11.25 + -7.75x + x2 = 0

Move the constant term to the right:

Add '-11.25' to each side of the equation.
11.25 + -7.75x + -11.25 + x2 = 0 + -11.25

Reorder the terms:
11.25 + -11.25 + -7.75x + x2 = 0 + -11.25

Combine like terms: 11.25 + -11.25 = 0.00
0.00 + -7.75x + x2 = 0 + -11.25
-7.75x + x2 = 0 + -11.25

Combine like terms: 0 + -11.25 = -11.25
-7.75x + x2 = -11.25

The x term is -7.75x.  Take half its coefficient (-3.875).
Square it (15.015625) and add it to both sides.

Add '15.015625' to each side of the equation.
-7.75x + 15.015625 + x2 = -11.25 + 15.015625

Reorder the terms:
15.015625 + -7.75x + x2 = -11.25 + 15.015625

Combine like terms: -11.25 + 15.015625 = 3.765625
15.015625 + -7.75x + x2 = 3.765625

Factor a perfect square on the left side:
(x + -3.875)(x + -3.875) = 3.765625

Calculate the square root of the right side: 1.940521837

Break this problem into two subproblems by setting 
(x + -3.875) equal to 1.940521837 and -1.940521837.

Subproblem 1

x + -3.875 = 1.940521837 Simplifying x + -3.875 = 1.940521837 Reorder the terms: -3.875 + x = 1.940521837 Solving -3.875 + x = 1.940521837 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + x = 1.940521837 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x = 1.940521837 + 3.875 x = 1.940521837 + 3.875 Combine like terms: 1.940521837 + 3.875 = 5.815521837 x = 5.815521837 Simplifying x = 5.815521837

Subproblem 2

x + -3.875 = -1.940521837 Simplifying x + -3.875 = -1.940521837 Reorder the terms: -3.875 + x = -1.940521837 Solving -3.875 + x = -1.940521837 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + x = -1.940521837 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x = -1.940521837 + 3.875 x = -1.940521837 + 3.875 Combine like terms: -1.940521837 + 3.875 = 1.934478163 x = 1.934478163 Simplifying x = 1.934478163

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.815521837, 1.934478163}

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