x^2+21x-54=0

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Solution for x^2+21x-54=0 equation:


Simplifying
x2 + 21x + -54 = 0

Reorder the terms:
-54 + 21x + x2 = 0

Solving
-54 + 21x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + 21x + 54 + x2 = 0 + 54

Reorder the terms:
-54 + 54 + 21x + x2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + 21x + x2 = 0 + 54
21x + x2 = 0 + 54

Combine like terms: 0 + 54 = 54
21x + x2 = 54

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 = 54 + 110.25

Reorder the terms:
110.25 + 21x + x2 = 54 + 110.25

Combine like terms: 54 + 110.25 = 164.25
110.25 + 21x + x2 = 164.25

Factor a perfect square on the left side:
(x + 10.5)(x + 10.5) = 164.25

Calculate the square root of the right side: 12.816005618

Break this problem into two subproblems by setting 
(x + 10.5) equal to 12.816005618 and -12.816005618.

Subproblem 1

x + 10.5 = 12.816005618 Simplifying x + 10.5 = 12.816005618 Reorder the terms: 10.5 + x = 12.816005618 Solving 10.5 + x = 12.816005618 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 = 12.816005618 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + x = 12.816005618 + -10.5 x = 12.816005618 + -10.5 Combine like terms: 12.816005618 + -10.5 = 2.316005618 x = 2.316005618 Simplifying x = 2.316005618

Subproblem 2

x + 10.5 = -12.816005618 Simplifying x + 10.5 = -12.816005618 Reorder the terms: 10.5 + x = -12.816005618 Solving 10.5 + x = -12.816005618 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 = -12.816005618 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + x = -12.816005618 + -10.5 x = -12.816005618 + -10.5 Combine like terms: -12.816005618 + -10.5 = -23.316005618 x = -23.316005618 Simplifying x = -23.316005618

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.316005618, -23.316005618}

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