52x=7x^2-67

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Solution for 52x=7x^2-67 equation:


Simplifying
52x = 7x2 + -67

Reorder the terms:
52x = -67 + 7x2

Solving
52x = -67 + 7x2

Solving for variable 'x'.

Reorder the terms:
67 + 52x + -7x2 = -67 + 7x2 + 67 + -7x2

Reorder the terms:
67 + 52x + -7x2 = -67 + 67 + 7x2 + -7x2

Combine like terms: -67 + 67 = 0
67 + 52x + -7x2 = 0 + 7x2 + -7x2
67 + 52x + -7x2 = 7x2 + -7x2

Combine like terms: 7x2 + -7x2 = 0
67 + 52x + -7x2 = 0

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

Divide each side by '-7'.
-9.571428571 + -7.428571429x + x2 = 0

Move the constant term to the right:

Add '9.571428571' to each side of the equation.
-9.571428571 + -7.428571429x + 9.571428571 + x2 = 0 + 9.571428571

Reorder the terms:
-9.571428571 + 9.571428571 + -7.428571429x + x2 = 0 + 9.571428571

Combine like terms: -9.571428571 + 9.571428571 = 0.000000000
0.000000000 + -7.428571429x + x2 = 0 + 9.571428571
-7.428571429x + x2 = 0 + 9.571428571

Combine like terms: 0 + 9.571428571 = 9.571428571
-7.428571429x + x2 = 9.571428571

The x term is -7.428571429x.  Take half its coefficient (-3.714285715).
Square it (13.79591837) and add it to both sides.

Add '13.79591837' to each side of the equation.
-7.428571429x + 13.79591837 + x2 = 9.571428571 + 13.79591837

Reorder the terms:
13.79591837 + -7.428571429x + x2 = 9.571428571 + 13.79591837

Combine like terms: 9.571428571 + 13.79591837 = 23.367346941
13.79591837 + -7.428571429x + x2 = 23.367346941

Factor a perfect square on the left side:
(x + -3.714285715)(x + -3.714285715) = 23.367346941

Calculate the square root of the right side: 4.833978376

Break this problem into two subproblems by setting 
(x + -3.714285715) equal to 4.833978376 and -4.833978376.

Subproblem 1

x + -3.714285715 = 4.833978376 Simplifying x + -3.714285715 = 4.833978376 Reorder the terms: -3.714285715 + x = 4.833978376 Solving -3.714285715 + x = 4.833978376 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.714285715' to each side of the equation. -3.714285715 + 3.714285715 + x = 4.833978376 + 3.714285715 Combine like terms: -3.714285715 + 3.714285715 = 0.000000000 0.000000000 + x = 4.833978376 + 3.714285715 x = 4.833978376 + 3.714285715 Combine like terms: 4.833978376 + 3.714285715 = 8.548264091 x = 8.548264091 Simplifying x = 8.548264091

Subproblem 2

x + -3.714285715 = -4.833978376 Simplifying x + -3.714285715 = -4.833978376 Reorder the terms: -3.714285715 + x = -4.833978376 Solving -3.714285715 + x = -4.833978376 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.714285715' to each side of the equation. -3.714285715 + 3.714285715 + x = -4.833978376 + 3.714285715 Combine like terms: -3.714285715 + 3.714285715 = 0.000000000 0.000000000 + x = -4.833978376 + 3.714285715 x = -4.833978376 + 3.714285715 Combine like terms: -4.833978376 + 3.714285715 = -1.119692661 x = -1.119692661 Simplifying x = -1.119692661

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.548264091, -1.119692661}

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