x^2-21x+84=0

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Solution for x^2-21x+84=0 equation:


Simplifying
x2 + -21x + 84 = 0

Reorder the terms:
84 + -21x + x2 = 0

Solving
84 + -21x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-84' to each side of the equation.
84 + -21x + -84 + x2 = 0 + -84

Reorder the terms:
84 + -84 + -21x + x2 = 0 + -84

Combine like terms: 84 + -84 = 0
0 + -21x + x2 = 0 + -84
-21x + x2 = 0 + -84

Combine like terms: 0 + -84 = -84
-21x + x2 = -84

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 = -84 + 110.25

Reorder the terms:
110.25 + -21x + x2 = -84 + 110.25

Combine like terms: -84 + 110.25 = 26.25
110.25 + -21x + x2 = 26.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 26.25

Calculate the square root of the right side: 5.123475383

Break this problem into two subproblems by setting 
(x + -10.5) equal to 5.123475383 and -5.123475383.

Subproblem 1

x + -10.5 = 5.123475383 Simplifying x + -10.5 = 5.123475383 Reorder the terms: -10.5 + x = 5.123475383 Solving -10.5 + x = 5.123475383 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 = 5.123475383 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = 5.123475383 + 10.5 x = 5.123475383 + 10.5 Combine like terms: 5.123475383 + 10.5 = 15.623475383 x = 15.623475383 Simplifying x = 15.623475383

Subproblem 2

x + -10.5 = -5.123475383 Simplifying x + -10.5 = -5.123475383 Reorder the terms: -10.5 + x = -5.123475383 Solving -10.5 + x = -5.123475383 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 = -5.123475383 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = -5.123475383 + 10.5 x = -5.123475383 + 10.5 Combine like terms: -5.123475383 + 10.5 = 5.376524617 x = 5.376524617 Simplifying x = 5.376524617

Solution

The solution to the problem is based on the solutions from the subproblems. x = {15.623475383, 5.376524617}

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