7x=2x^2+1

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Solution for 7x=2x^2+1 equation:


Simplifying
7x = 2x2 + 1

Reorder the terms:
7x = 1 + 2x2

Solving
7x = 1 + 2x2

Solving for variable 'x'.

Reorder the terms:
-1 + 7x + -2x2 = 1 + 2x2 + -1 + -2x2

Reorder the terms:
-1 + 7x + -2x2 = 1 + -1 + 2x2 + -2x2

Combine like terms: 1 + -1 = 0
-1 + 7x + -2x2 = 0 + 2x2 + -2x2
-1 + 7x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
-1 + 7x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
0.5 + -3.5x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -3.5x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -3.5x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -3.5x + x2 = 0 + -0.5
-3.5x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-3.5x + x2 = -0.5

The x term is -3.5x.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5x + 3.0625 + x2 = -0.5 + 3.0625

Reorder the terms:
3.0625 + -3.5x + x2 = -0.5 + 3.0625

Combine like terms: -0.5 + 3.0625 = 2.5625
3.0625 + -3.5x + x2 = 2.5625

Factor a perfect square on the left side:
(x + -1.75)(x + -1.75) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(x + -1.75) equal to 1.600781059 and -1.600781059.

Subproblem 1

x + -1.75 = 1.600781059 Simplifying x + -1.75 = 1.600781059 Reorder the terms: -1.75 + x = 1.600781059 Solving -1.75 + x = 1.600781059 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x = 1.600781059 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x = 1.600781059 + 1.75 x = 1.600781059 + 1.75 Combine like terms: 1.600781059 + 1.75 = 3.350781059 x = 3.350781059 Simplifying x = 3.350781059

Subproblem 2

x + -1.75 = -1.600781059 Simplifying x + -1.75 = -1.600781059 Reorder the terms: -1.75 + x = -1.600781059 Solving -1.75 + x = -1.600781059 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x = -1.600781059 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x = -1.600781059 + 1.75 x = -1.600781059 + 1.75 Combine like terms: -1.600781059 + 1.75 = 0.149218941 x = 0.149218941 Simplifying x = 0.149218941

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.350781059, 0.149218941}

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