x^2-3x-52=0

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Solution for x^2-3x-52=0 equation:


Simplifying
x2 + -3x + -52 = 0

Reorder the terms:
-52 + -3x + x2 = 0

Solving
-52 + -3x + 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 + -3x + 52 + x2 = 0 + 52

Reorder the terms:
-52 + 52 + -3x + x2 = 0 + 52

Combine like terms: -52 + 52 = 0
0 + -3x + x2 = 0 + 52
-3x + x2 = 0 + 52

Combine like terms: 0 + 52 = 52
-3x + x2 = 52

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 52 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 52 + 2.25

Combine like terms: 52 + 2.25 = 54.25
2.25 + -3x + x2 = 54.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 54.25

Calculate the square root of the right side: 7.365459931

Break this problem into two subproblems by setting 
(x + -1.5) equal to 7.365459931 and -7.365459931.

Subproblem 1

x + -1.5 = 7.365459931 Simplifying x + -1.5 = 7.365459931 Reorder the terms: -1.5 + x = 7.365459931 Solving -1.5 + x = 7.365459931 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 7.365459931 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 7.365459931 + 1.5 x = 7.365459931 + 1.5 Combine like terms: 7.365459931 + 1.5 = 8.865459931 x = 8.865459931 Simplifying x = 8.865459931

Subproblem 2

x + -1.5 = -7.365459931 Simplifying x + -1.5 = -7.365459931 Reorder the terms: -1.5 + x = -7.365459931 Solving -1.5 + x = -7.365459931 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -7.365459931 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -7.365459931 + 1.5 x = -7.365459931 + 1.5 Combine like terms: -7.365459931 + 1.5 = -5.865459931 x = -5.865459931 Simplifying x = -5.865459931

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.865459931, -5.865459931}

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