90-x+10=2x^2+35

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Solution for 90-x+10=2x^2+35 equation:


Simplifying
90 + -1x + 10 = 2x2 + 35

Reorder the terms:
90 + 10 + -1x = 2x2 + 35

Combine like terms: 90 + 10 = 100
100 + -1x = 2x2 + 35

Reorder the terms:
100 + -1x = 35 + 2x2

Solving
100 + -1x = 35 + 2x2

Solving for variable 'x'.

Reorder the terms:
100 + -35 + -1x + -2x2 = 35 + 2x2 + -35 + -2x2

Combine like terms: 100 + -35 = 65
65 + -1x + -2x2 = 35 + 2x2 + -35 + -2x2

Reorder the terms:
65 + -1x + -2x2 = 35 + -35 + 2x2 + -2x2

Combine like terms: 35 + -35 = 0
65 + -1x + -2x2 = 0 + 2x2 + -2x2
65 + -1x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
65 + -1x + -2x2 = 0

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

Divide each side by '-2'.
-32.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '32.5' to each side of the equation.
-32.5 + 0.5x + 32.5 + x2 = 0 + 32.5

Reorder the terms:
-32.5 + 32.5 + 0.5x + x2 = 0 + 32.5

Combine like terms: -32.5 + 32.5 = 0.0
0.0 + 0.5x + x2 = 0 + 32.5
0.5x + x2 = 0 + 32.5

Combine like terms: 0 + 32.5 = 32.5
0.5x + x2 = 32.5

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 32.5 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 32.5 + 0.0625

Combine like terms: 32.5 + 0.0625 = 32.5625
0.0625 + 0.5x + x2 = 32.5625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 32.5625

Calculate the square root of the right side: 5.706356105

Break this problem into two subproblems by setting 
(x + 0.25) equal to 5.706356105 and -5.706356105.

Subproblem 1

x + 0.25 = 5.706356105 Simplifying x + 0.25 = 5.706356105 Reorder the terms: 0.25 + x = 5.706356105 Solving 0.25 + x = 5.706356105 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 5.706356105 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.706356105 + -0.25 x = 5.706356105 + -0.25 Combine like terms: 5.706356105 + -0.25 = 5.456356105 x = 5.456356105 Simplifying x = 5.456356105

Subproblem 2

x + 0.25 = -5.706356105 Simplifying x + 0.25 = -5.706356105 Reorder the terms: 0.25 + x = -5.706356105 Solving 0.25 + x = -5.706356105 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -5.706356105 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.706356105 + -0.25 x = -5.706356105 + -0.25 Combine like terms: -5.706356105 + -0.25 = -5.956356105 x = -5.956356105 Simplifying x = -5.956356105

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.456356105, -5.956356105}

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