x^2+5x=52

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Solution for x^2+5x=52 equation:


Simplifying
x2 + 5x = 52

Reorder the terms:
5x + x2 = 52

Solving
5x + x2 = 52

Solving for variable 'x'.

Reorder the terms:
-52 + 5x + x2 = 52 + -52

Combine like terms: 52 + -52 = 0
-52 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '52' to each side of the equation.
-52 + 5x + 52 + x2 = 0 + 52

Reorder the terms:
-52 + 52 + 5x + x2 = 0 + 52

Combine like terms: -52 + 52 = 0
0 + 5x + x2 = 0 + 52
5x + x2 = 0 + 52

Combine like terms: 0 + 52 = 52
5x + x2 = 52

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 52 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 52 + 6.25

Combine like terms: 52 + 6.25 = 58.25
6.25 + 5x + x2 = 58.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 58.25

Calculate the square root of the right side: 7.632168761

Break this problem into two subproblems by setting 
(x + 2.5) equal to 7.632168761 and -7.632168761.

Subproblem 1

x + 2.5 = 7.632168761 Simplifying x + 2.5 = 7.632168761 Reorder the terms: 2.5 + x = 7.632168761 Solving 2.5 + x = 7.632168761 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 7.632168761 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 7.632168761 + -2.5 x = 7.632168761 + -2.5 Combine like terms: 7.632168761 + -2.5 = 5.132168761 x = 5.132168761 Simplifying x = 5.132168761

Subproblem 2

x + 2.5 = -7.632168761 Simplifying x + 2.5 = -7.632168761 Reorder the terms: 2.5 + x = -7.632168761 Solving 2.5 + x = -7.632168761 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -7.632168761 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -7.632168761 + -2.5 x = -7.632168761 + -2.5 Combine like terms: -7.632168761 + -2.5 = -10.132168761 x = -10.132168761 Simplifying x = -10.132168761

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.132168761, -10.132168761}

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