7x^2+84-53x=0

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Solution for 7x^2+84-53x=0 equation:


Simplifying
7x2 + 84 + -53x = 0

Reorder the terms:
84 + -53x + 7x2 = 0

Solving
84 + -53x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
12 + -7.571428571x + x2 = 0

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + -7.571428571x + -12 + x2 = 0 + -12

Reorder the terms:
12 + -12 + -7.571428571x + x2 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + -7.571428571x + x2 = 0 + -12
-7.571428571x + x2 = 0 + -12

Combine like terms: 0 + -12 = -12
-7.571428571x + x2 = -12

The x term is -7.571428571x.  Take half its coefficient (-3.785714286).
Square it (14.33163266) and add it to both sides.

Add '14.33163266' to each side of the equation.
-7.571428571x + 14.33163266 + x2 = -12 + 14.33163266

Reorder the terms:
14.33163266 + -7.571428571x + x2 = -12 + 14.33163266

Combine like terms: -12 + 14.33163266 = 2.33163266
14.33163266 + -7.571428571x + x2 = 2.33163266

Factor a perfect square on the left side:
(x + -3.785714286)(x + -3.785714286) = 2.33163266

Calculate the square root of the right side: 1.526968454

Break this problem into two subproblems by setting 
(x + -3.785714286) equal to 1.526968454 and -1.526968454.

Subproblem 1

x + -3.785714286 = 1.526968454 Simplifying x + -3.785714286 = 1.526968454 Reorder the terms: -3.785714286 + x = 1.526968454 Solving -3.785714286 + x = 1.526968454 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.785714286' to each side of the equation. -3.785714286 + 3.785714286 + x = 1.526968454 + 3.785714286 Combine like terms: -3.785714286 + 3.785714286 = 0.000000000 0.000000000 + x = 1.526968454 + 3.785714286 x = 1.526968454 + 3.785714286 Combine like terms: 1.526968454 + 3.785714286 = 5.31268274 x = 5.31268274 Simplifying x = 5.31268274

Subproblem 2

x + -3.785714286 = -1.526968454 Simplifying x + -3.785714286 = -1.526968454 Reorder the terms: -3.785714286 + x = -1.526968454 Solving -3.785714286 + x = -1.526968454 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.785714286' to each side of the equation. -3.785714286 + 3.785714286 + x = -1.526968454 + 3.785714286 Combine like terms: -3.785714286 + 3.785714286 = 0.000000000 0.000000000 + x = -1.526968454 + 3.785714286 x = -1.526968454 + 3.785714286 Combine like terms: -1.526968454 + 3.785714286 = 2.258745832 x = 2.258745832 Simplifying x = 2.258745832

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.31268274, 2.258745832}

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