x^2-6.5x-52=0

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Solution for x^2-6.5x-52=0 equation:


Simplifying
x2 + -6.5x + -52 = 0

Reorder the terms:
-52 + -6.5x + x2 = 0

Solving
-52 + -6.5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '52' to each side of the equation.
-52 + -6.5x + 52 + x2 = 0 + 52

Reorder the terms:
-52 + 52 + -6.5x + x2 = 0 + 52

Combine like terms: -52 + 52 = 0
0 + -6.5x + x2 = 0 + 52
-6.5x + x2 = 0 + 52

Combine like terms: 0 + 52 = 52
-6.5x + x2 = 52

The x term is -6.5x.  Take half its coefficient (-3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
-6.5x + 10.5625 + x2 = 52 + 10.5625

Reorder the terms:
10.5625 + -6.5x + x2 = 52 + 10.5625

Combine like terms: 52 + 10.5625 = 62.5625
10.5625 + -6.5x + x2 = 62.5625

Factor a perfect square on the left side:
(x + -3.25)(x + -3.25) = 62.5625

Calculate the square root of the right side: 7.90964601

Break this problem into two subproblems by setting 
(x + -3.25) equal to 7.90964601 and -7.90964601.

Subproblem 1

x + -3.25 = 7.90964601 Simplifying x + -3.25 = 7.90964601 Reorder the terms: -3.25 + x = 7.90964601 Solving -3.25 + x = 7.90964601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = 7.90964601 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = 7.90964601 + 3.25 x = 7.90964601 + 3.25 Combine like terms: 7.90964601 + 3.25 = 11.15964601 x = 11.15964601 Simplifying x = 11.15964601

Subproblem 2

x + -3.25 = -7.90964601 Simplifying x + -3.25 = -7.90964601 Reorder the terms: -3.25 + x = -7.90964601 Solving -3.25 + x = -7.90964601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = -7.90964601 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = -7.90964601 + 3.25 x = -7.90964601 + 3.25 Combine like terms: -7.90964601 + 3.25 = -4.65964601 x = -4.65964601 Simplifying x = -4.65964601

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11.15964601, -4.65964601}

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