2x^2=3x+23

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Solution for 2x^2=3x+23 equation:


Simplifying
2x2 = 3x + 23

Reorder the terms:
2x2 = 23 + 3x

Solving
2x2 = 23 + 3x

Solving for variable 'x'.

Reorder the terms:
-23 + -3x + 2x2 = 23 + 3x + -23 + -3x

Reorder the terms:
-23 + -3x + 2x2 = 23 + -23 + 3x + -3x

Combine like terms: 23 + -23 = 0
-23 + -3x + 2x2 = 0 + 3x + -3x
-23 + -3x + 2x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
-23 + -3x + 2x2 = 0

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

Divide each side by '2'.
-11.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '11.5' to each side of the equation.
-11.5 + -1.5x + 11.5 + x2 = 0 + 11.5

Reorder the terms:
-11.5 + 11.5 + -1.5x + x2 = 0 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + -1.5x + x2 = 0 + 11.5
-1.5x + x2 = 0 + 11.5

Combine like terms: 0 + 11.5 = 11.5
-1.5x + x2 = 11.5

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 11.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 11.5 + 0.5625

Combine like terms: 11.5 + 0.5625 = 12.0625
0.5625 + -1.5x + x2 = 12.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 12.0625

Calculate the square root of the right side: 3.473110997

Break this problem into two subproblems by setting 
(x + -0.75) equal to 3.473110997 and -3.473110997.

Subproblem 1

x + -0.75 = 3.473110997 Simplifying x + -0.75 = 3.473110997 Reorder the terms: -0.75 + x = 3.473110997 Solving -0.75 + x = 3.473110997 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 3.473110997 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 3.473110997 + 0.75 x = 3.473110997 + 0.75 Combine like terms: 3.473110997 + 0.75 = 4.223110997 x = 4.223110997 Simplifying x = 4.223110997

Subproblem 2

x + -0.75 = -3.473110997 Simplifying x + -0.75 = -3.473110997 Reorder the terms: -0.75 + x = -3.473110997 Solving -0.75 + x = -3.473110997 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -3.473110997 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -3.473110997 + 0.75 x = -3.473110997 + 0.75 Combine like terms: -3.473110997 + 0.75 = -2.723110997 x = -2.723110997 Simplifying x = -2.723110997

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.223110997, -2.723110997}

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