x^2-21x=-32

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Solution for x^2-21x=-32 equation:


Simplifying
x2 + -21x = -32

Reorder the terms:
-21x + x2 = -32

Solving
-21x + x2 = -32

Solving for variable 'x'.

Reorder the terms:
32 + -21x + x2 = -32 + 32

Combine like terms: -32 + 32 = 0
32 + -21x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-32' to each side of the equation.
32 + -21x + -32 + x2 = 0 + -32

Reorder the terms:
32 + -32 + -21x + x2 = 0 + -32

Combine like terms: 32 + -32 = 0
0 + -21x + x2 = 0 + -32
-21x + x2 = 0 + -32

Combine like terms: 0 + -32 = -32
-21x + x2 = -32

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 = -32 + 110.25

Reorder the terms:
110.25 + -21x + x2 = -32 + 110.25

Combine like terms: -32 + 110.25 = 78.25
110.25 + -21x + x2 = 78.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 78.25

Calculate the square root of the right side: 8.845903006

Break this problem into two subproblems by setting 
(x + -10.5) equal to 8.845903006 and -8.845903006.

Subproblem 1

x + -10.5 = 8.845903006 Simplifying x + -10.5 = 8.845903006 Reorder the terms: -10.5 + x = 8.845903006 Solving -10.5 + x = 8.845903006 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 = 8.845903006 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = 8.845903006 + 10.5 x = 8.845903006 + 10.5 Combine like terms: 8.845903006 + 10.5 = 19.345903006 x = 19.345903006 Simplifying x = 19.345903006

Subproblem 2

x + -10.5 = -8.845903006 Simplifying x + -10.5 = -8.845903006 Reorder the terms: -10.5 + x = -8.845903006 Solving -10.5 + x = -8.845903006 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 = -8.845903006 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = -8.845903006 + 10.5 x = -8.845903006 + 10.5 Combine like terms: -8.845903006 + 10.5 = 1.654096994 x = 1.654096994 Simplifying x = 1.654096994

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.345903006, 1.654096994}

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