68x-2x^2=21x+97

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Solution for 68x-2x^2=21x+97 equation:


Simplifying
68x + -2x2 = 21x + 97

Reorder the terms:
68x + -2x2 = 97 + 21x

Solving
68x + -2x2 = 97 + 21x

Solving for variable 'x'.

Reorder the terms:
-97 + 68x + -21x + -2x2 = 97 + 21x + -97 + -21x

Combine like terms: 68x + -21x = 47x
-97 + 47x + -2x2 = 97 + 21x + -97 + -21x

Reorder the terms:
-97 + 47x + -2x2 = 97 + -97 + 21x + -21x

Combine like terms: 97 + -97 = 0
-97 + 47x + -2x2 = 0 + 21x + -21x
-97 + 47x + -2x2 = 21x + -21x

Combine like terms: 21x + -21x = 0
-97 + 47x + -2x2 = 0

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

Divide each side by '-2'.
48.5 + -23.5x + x2 = 0

Move the constant term to the right:

Add '-48.5' to each side of the equation.
48.5 + -23.5x + -48.5 + x2 = 0 + -48.5

Reorder the terms:
48.5 + -48.5 + -23.5x + x2 = 0 + -48.5

Combine like terms: 48.5 + -48.5 = 0.0
0.0 + -23.5x + x2 = 0 + -48.5
-23.5x + x2 = 0 + -48.5

Combine like terms: 0 + -48.5 = -48.5
-23.5x + x2 = -48.5

The x term is -23.5x.  Take half its coefficient (-11.75).
Square it (138.0625) and add it to both sides.

Add '138.0625' to each side of the equation.
-23.5x + 138.0625 + x2 = -48.5 + 138.0625

Reorder the terms:
138.0625 + -23.5x + x2 = -48.5 + 138.0625

Combine like terms: -48.5 + 138.0625 = 89.5625
138.0625 + -23.5x + x2 = 89.5625

Factor a perfect square on the left side:
(x + -11.75)(x + -11.75) = 89.5625

Calculate the square root of the right side: 9.463746615

Break this problem into two subproblems by setting 
(x + -11.75) equal to 9.463746615 and -9.463746615.

Subproblem 1

x + -11.75 = 9.463746615 Simplifying x + -11.75 = 9.463746615 Reorder the terms: -11.75 + x = 9.463746615 Solving -11.75 + x = 9.463746615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.75' to each side of the equation. -11.75 + 11.75 + x = 9.463746615 + 11.75 Combine like terms: -11.75 + 11.75 = 0.00 0.00 + x = 9.463746615 + 11.75 x = 9.463746615 + 11.75 Combine like terms: 9.463746615 + 11.75 = 21.213746615 x = 21.213746615 Simplifying x = 21.213746615

Subproblem 2

x + -11.75 = -9.463746615 Simplifying x + -11.75 = -9.463746615 Reorder the terms: -11.75 + x = -9.463746615 Solving -11.75 + x = -9.463746615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.75' to each side of the equation. -11.75 + 11.75 + x = -9.463746615 + 11.75 Combine like terms: -11.75 + 11.75 = 0.00 0.00 + x = -9.463746615 + 11.75 x = -9.463746615 + 11.75 Combine like terms: -9.463746615 + 11.75 = 2.286253385 x = 2.286253385 Simplifying x = 2.286253385

Solution

The solution to the problem is based on the solutions from the subproblems. x = {21.213746615, 2.286253385}

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