90-x^2+4=x+30

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Solution for 90-x^2+4=x+30 equation:


Simplifying
90 + -1x2 + 4 = x + 30

Reorder the terms:
90 + 4 + -1x2 = x + 30

Combine like terms: 90 + 4 = 94
94 + -1x2 = x + 30

Reorder the terms:
94 + -1x2 = 30 + x

Solving
94 + -1x2 = 30 + x

Solving for variable 'x'.

Reorder the terms:
94 + -30 + -1x + -1x2 = 30 + x + -30 + -1x

Combine like terms: 94 + -30 = 64
64 + -1x + -1x2 = 30 + x + -30 + -1x

Reorder the terms:
64 + -1x + -1x2 = 30 + -30 + x + -1x

Combine like terms: 30 + -30 = 0
64 + -1x + -1x2 = 0 + x + -1x
64 + -1x + -1x2 = x + -1x

Combine like terms: x + -1x = 0
64 + -1x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-64 + x + x2 = 0

Move the constant term to the right:

Add '64' to each side of the equation.
-64 + x + 64 + x2 = 0 + 64

Reorder the terms:
-64 + 64 + x + x2 = 0 + 64

Combine like terms: -64 + 64 = 0
0 + x + x2 = 0 + 64
x + x2 = 0 + 64

Combine like terms: 0 + 64 = 64
x + x2 = 64

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

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 64 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 64 + 0.25

Combine like terms: 64 + 0.25 = 64.25
1.25 + x2 = 64.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 64.25

Calculate the square root of the right side: 8.015609771

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.015609771 and -8.015609771.

Subproblem 1

x + 0.5 = 8.015609771 Simplifying x + 0.5 = 8.015609771 Reorder the terms: 0.5 + x = 8.015609771 Solving 0.5 + x = 8.015609771 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 8.015609771 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.015609771 + -0.5 x = 8.015609771 + -0.5 Combine like terms: 8.015609771 + -0.5 = 7.515609771 x = 7.515609771 Simplifying x = 7.515609771

Subproblem 2

x + 0.5 = -8.015609771 Simplifying x + 0.5 = -8.015609771 Reorder the terms: 0.5 + x = -8.015609771 Solving 0.5 + x = -8.015609771 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -8.015609771 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.015609771 + -0.5 x = -8.015609771 + -0.5 Combine like terms: -8.015609771 + -0.5 = -8.515609771 x = -8.515609771 Simplifying x = -8.515609771

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.515609771, -8.515609771}

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