7x-4=x^2+10x-53

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Solution for 7x-4=x^2+10x-53 equation:


Simplifying
7x + -4 = x2 + 10x + -53

Reorder the terms:
-4 + 7x = x2 + 10x + -53

Reorder the terms:
-4 + 7x = -53 + 10x + x2

Solving
-4 + 7x = -53 + 10x + x2

Solving for variable 'x'.

Reorder the terms:
-4 + 53 + 7x + -10x + -1x2 = -53 + 10x + x2 + 53 + -10x + -1x2

Combine like terms: -4 + 53 = 49
49 + 7x + -10x + -1x2 = -53 + 10x + x2 + 53 + -10x + -1x2

Combine like terms: 7x + -10x = -3x
49 + -3x + -1x2 = -53 + 10x + x2 + 53 + -10x + -1x2

Reorder the terms:
49 + -3x + -1x2 = -53 + 53 + 10x + -10x + x2 + -1x2

Combine like terms: -53 + 53 = 0
49 + -3x + -1x2 = 0 + 10x + -10x + x2 + -1x2
49 + -3x + -1x2 = 10x + -10x + x2 + -1x2

Combine like terms: 10x + -10x = 0
49 + -3x + -1x2 = 0 + x2 + -1x2
49 + -3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
49 + -3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-49 + 3x + x2 = 0

Move the constant term to the right:

Add '49' to each side of the equation.
-49 + 3x + 49 + x2 = 0 + 49

Reorder the terms:
-49 + 49 + 3x + x2 = 0 + 49

Combine like terms: -49 + 49 = 0
0 + 3x + x2 = 0 + 49
3x + x2 = 0 + 49

Combine like terms: 0 + 49 = 49
3x + x2 = 49

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 = 49 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 49 + 2.25

Combine like terms: 49 + 2.25 = 51.25
2.25 + 3x + x2 = 51.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 51.25

Calculate the square root of the right side: 7.158910532

Break this problem into two subproblems by setting 
(x + 1.5) equal to 7.158910532 and -7.158910532.

Subproblem 1

x + 1.5 = 7.158910532 Simplifying x + 1.5 = 7.158910532 Reorder the terms: 1.5 + x = 7.158910532 Solving 1.5 + x = 7.158910532 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 = 7.158910532 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 7.158910532 + -1.5 x = 7.158910532 + -1.5 Combine like terms: 7.158910532 + -1.5 = 5.658910532 x = 5.658910532 Simplifying x = 5.658910532

Subproblem 2

x + 1.5 = -7.158910532 Simplifying x + 1.5 = -7.158910532 Reorder the terms: 1.5 + x = -7.158910532 Solving 1.5 + x = -7.158910532 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 = -7.158910532 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -7.158910532 + -1.5 x = -7.158910532 + -1.5 Combine like terms: -7.158910532 + -1.5 = -8.658910532 x = -8.658910532 Simplifying x = -8.658910532

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.658910532, -8.658910532}

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