2x^2-2x-21=0

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


Simplifying
2x2 + -2x + -21 = 0

Reorder the terms:
-21 + -2x + 2x2 = 0

Solving
-21 + -2x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-10.5 + -1x + x2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + -1x + 10.5 + x2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + -1x + x2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + -1x + x2 = 0 + 10.5
-1x + x2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
-1x + x2 = 10.5

The x term is -1x.  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.
-1x + 0.25 + x2 = 10.5 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 10.5 + 0.25

Combine like terms: 10.5 + 0.25 = 10.75
0.25 + -1x + x2 = 10.75

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 10.75

Calculate the square root of the right side: 3.278719262

Break this problem into two subproblems by setting 
(x + -0.5) equal to 3.278719262 and -3.278719262.

Subproblem 1

x + -0.5 = 3.278719262 Simplifying x + -0.5 = 3.278719262 Reorder the terms: -0.5 + x = 3.278719262 Solving -0.5 + x = 3.278719262 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 = 3.278719262 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 3.278719262 + 0.5 x = 3.278719262 + 0.5 Combine like terms: 3.278719262 + 0.5 = 3.778719262 x = 3.778719262 Simplifying x = 3.778719262

Subproblem 2

x + -0.5 = -3.278719262 Simplifying x + -0.5 = -3.278719262 Reorder the terms: -0.5 + x = -3.278719262 Solving -0.5 + x = -3.278719262 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 = -3.278719262 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -3.278719262 + 0.5 x = -3.278719262 + 0.5 Combine like terms: -3.278719262 + 0.5 = -2.778719262 x = -2.778719262 Simplifying x = -2.778719262

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.778719262, -2.778719262}

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