5x^2=12x+31

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Solution for 5x^2=12x+31 equation:


Simplifying
5x2 = 12x + 31

Reorder the terms:
5x2 = 31 + 12x

Solving
5x2 = 31 + 12x

Solving for variable 'x'.

Reorder the terms:
-31 + -12x + 5x2 = 31 + 12x + -31 + -12x

Reorder the terms:
-31 + -12x + 5x2 = 31 + -31 + 12x + -12x

Combine like terms: 31 + -31 = 0
-31 + -12x + 5x2 = 0 + 12x + -12x
-31 + -12x + 5x2 = 12x + -12x

Combine like terms: 12x + -12x = 0
-31 + -12x + 5x2 = 0

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

Divide each side by '5'.
-6.2 + -2.4x + x2 = 0

Move the constant term to the right:

Add '6.2' to each side of the equation.
-6.2 + -2.4x + 6.2 + x2 = 0 + 6.2

Reorder the terms:
-6.2 + 6.2 + -2.4x + x2 = 0 + 6.2

Combine like terms: -6.2 + 6.2 = 0.0
0.0 + -2.4x + x2 = 0 + 6.2
-2.4x + x2 = 0 + 6.2

Combine like terms: 0 + 6.2 = 6.2
-2.4x + x2 = 6.2

The x term is -2.4x.  Take half its coefficient (-1.2).
Square it (1.44) and add it to both sides.

Add '1.44' to each side of the equation.
-2.4x + 1.44 + x2 = 6.2 + 1.44

Reorder the terms:
1.44 + -2.4x + x2 = 6.2 + 1.44

Combine like terms: 6.2 + 1.44 = 7.64
1.44 + -2.4x + x2 = 7.64

Factor a perfect square on the left side:
(x + -1.2)(x + -1.2) = 7.64

Calculate the square root of the right side: 2.764054992

Break this problem into two subproblems by setting 
(x + -1.2) equal to 2.764054992 and -2.764054992.

Subproblem 1

x + -1.2 = 2.764054992 Simplifying x + -1.2 = 2.764054992 Reorder the terms: -1.2 + x = 2.764054992 Solving -1.2 + x = 2.764054992 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + x = 2.764054992 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + x = 2.764054992 + 1.2 x = 2.764054992 + 1.2 Combine like terms: 2.764054992 + 1.2 = 3.964054992 x = 3.964054992 Simplifying x = 3.964054992

Subproblem 2

x + -1.2 = -2.764054992 Simplifying x + -1.2 = -2.764054992 Reorder the terms: -1.2 + x = -2.764054992 Solving -1.2 + x = -2.764054992 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + x = -2.764054992 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + x = -2.764054992 + 1.2 x = -2.764054992 + 1.2 Combine like terms: -2.764054992 + 1.2 = -1.564054992 x = -1.564054992 Simplifying x = -1.564054992

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.964054992, -1.564054992}

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