21x-x^2-30=0

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Solution for 21x-x^2-30=0 equation:


Simplifying
21x + -1x2 + -30 = 0

Reorder the terms:
-30 + 21x + -1x2 = 0

Solving
-30 + 21x + -1x2 = 0

Solving for variable 'x'.

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

Divide each side by '-1'.
30 + -21x + x2 = 0

Move the constant term to the right:

Add '-30' to each side of the equation.
30 + -21x + -30 + x2 = 0 + -30

Reorder the terms:
30 + -30 + -21x + x2 = 0 + -30

Combine like terms: 30 + -30 = 0
0 + -21x + x2 = 0 + -30
-21x + x2 = 0 + -30

Combine like terms: 0 + -30 = -30
-21x + x2 = -30

The x term is -21x.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21x + 110.25 + x2 = -30 + 110.25

Reorder the terms:
110.25 + -21x + x2 = -30 + 110.25

Combine like terms: -30 + 110.25 = 80.25
110.25 + -21x + x2 = 80.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 80.25

Calculate the square root of the right side: 8.958236434

Break this problem into two subproblems by setting 
(x + -10.5) equal to 8.958236434 and -8.958236434.

Subproblem 1

x + -10.5 = 8.958236434 Simplifying x + -10.5 = 8.958236434 Reorder the terms: -10.5 + x = 8.958236434 Solving -10.5 + x = 8.958236434 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = 8.958236434 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = 8.958236434 + 10.5 x = 8.958236434 + 10.5 Combine like terms: 8.958236434 + 10.5 = 19.458236434 x = 19.458236434 Simplifying x = 19.458236434

Subproblem 2

x + -10.5 = -8.958236434 Simplifying x + -10.5 = -8.958236434 Reorder the terms: -10.5 + x = -8.958236434 Solving -10.5 + x = -8.958236434 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = -8.958236434 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = -8.958236434 + 10.5 x = -8.958236434 + 10.5 Combine like terms: -8.958236434 + 10.5 = 1.541763566 x = 1.541763566 Simplifying x = 1.541763566

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.458236434, 1.541763566}

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