x^2+3x-53=0

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Solution for x^2+3x-53=0 equation:


Simplifying
x2 + 3x + -53 = 0

Reorder the terms:
-53 + 3x + x2 = 0

Solving
-53 + 3x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '53' to each side of the equation.
-53 + 3x + 53 + x2 = 0 + 53

Reorder the terms:
-53 + 53 + 3x + x2 = 0 + 53

Combine like terms: -53 + 53 = 0
0 + 3x + x2 = 0 + 53
3x + x2 = 0 + 53

Combine like terms: 0 + 53 = 53
3x + x2 = 53

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 = 53 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 53 + 2.25

Combine like terms: 53 + 2.25 = 55.25
2.25 + 3x + x2 = 55.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 55.25

Calculate the square root of the right side: 7.433034374

Break this problem into two subproblems by setting 
(x + 1.5) equal to 7.433034374 and -7.433034374.

Subproblem 1

x + 1.5 = 7.433034374 Simplifying x + 1.5 = 7.433034374 Reorder the terms: 1.5 + x = 7.433034374 Solving 1.5 + x = 7.433034374 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 = 7.433034374 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 7.433034374 + -1.5 x = 7.433034374 + -1.5 Combine like terms: 7.433034374 + -1.5 = 5.933034374 x = 5.933034374 Simplifying x = 5.933034374

Subproblem 2

x + 1.5 = -7.433034374 Simplifying x + 1.5 = -7.433034374 Reorder the terms: 1.5 + x = -7.433034374 Solving 1.5 + x = -7.433034374 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 = -7.433034374 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -7.433034374 + -1.5 x = -7.433034374 + -1.5 Combine like terms: -7.433034374 + -1.5 = -8.933034374 x = -8.933034374 Simplifying x = -8.933034374

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.933034374, -8.933034374}

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