4x^2-31x+15=0

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Solution for 4x^2-31x+15=0 equation:


Simplifying
4x2 + -31x + 15 = 0

Reorder the terms:
15 + -31x + 4x2 = 0

Solving
15 + -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'.
3.75 + -7.75x + x2 = 0

Move the constant term to the right:

Add '-3.75' to each side of the equation.
3.75 + -7.75x + -3.75 + x2 = 0 + -3.75

Reorder the terms:
3.75 + -3.75 + -7.75x + x2 = 0 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + -7.75x + x2 = 0 + -3.75
-7.75x + x2 = 0 + -3.75

Combine like terms: 0 + -3.75 = -3.75
-7.75x + x2 = -3.75

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 = -3.75 + 15.015625

Reorder the terms:
15.015625 + -7.75x + x2 = -3.75 + 15.015625

Combine like terms: -3.75 + 15.015625 = 11.265625
15.015625 + -7.75x + x2 = 11.265625

Factor a perfect square on the left side:
(x + -3.875)(x + -3.875) = 11.265625

Calculate the square root of the right side: 3.356430396

Break this problem into two subproblems by setting 
(x + -3.875) equal to 3.356430396 and -3.356430396.

Subproblem 1

x + -3.875 = 3.356430396 Simplifying x + -3.875 = 3.356430396 Reorder the terms: -3.875 + x = 3.356430396 Solving -3.875 + x = 3.356430396 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 = 3.356430396 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x = 3.356430396 + 3.875 x = 3.356430396 + 3.875 Combine like terms: 3.356430396 + 3.875 = 7.231430396 x = 7.231430396 Simplifying x = 7.231430396

Subproblem 2

x + -3.875 = -3.356430396 Simplifying x + -3.875 = -3.356430396 Reorder the terms: -3.875 + x = -3.356430396 Solving -3.875 + x = -3.356430396 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 = -3.356430396 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x = -3.356430396 + 3.875 x = -3.356430396 + 3.875 Combine like terms: -3.356430396 + 3.875 = 0.518569604 x = 0.518569604 Simplifying x = 0.518569604

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.231430396, 0.518569604}

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