x+7x^2=10

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Solution for x+7x^2=10 equation:


Simplifying
x + 7x2 = 10

Solving
x + 7x2 = 10

Solving for variable 'x'.

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

Combine like terms: 10 + -10 = 0
-10 + x + 7x2 = 0

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

Divide each side by '7'.
-1.428571429 + 0.1428571429x + x2 = 0

Move the constant term to the right:

Add '1.428571429' to each side of the equation.
-1.428571429 + 0.1428571429x + 1.428571429 + x2 = 0 + 1.428571429

Reorder the terms:
-1.428571429 + 1.428571429 + 0.1428571429x + x2 = 0 + 1.428571429

Combine like terms: -1.428571429 + 1.428571429 = 0.000000000
0.000000000 + 0.1428571429x + x2 = 0 + 1.428571429
0.1428571429x + x2 = 0 + 1.428571429

Combine like terms: 0 + 1.428571429 = 1.428571429
0.1428571429x + x2 = 1.428571429

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1428571429x + 0.25 + x2 = 1.428571429 + 0.25

Reorder the terms:
0.25 + 0.1428571429x + x2 = 1.428571429 + 0.25

Combine like terms: 1.428571429 + 0.25 = 1.678571429
0.25 + 0.1428571429x + x2 = 1.678571429

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1.678571429

Calculate the square root of the right side: 1.295596939

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.295596939 and -1.295596939.

Subproblem 1

x + 0.5 = 1.295596939 Simplifying x + 0.5 = 1.295596939 Reorder the terms: 0.5 + x = 1.295596939 Solving 0.5 + x = 1.295596939 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 1.295596939 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.295596939 + -0.5 x = 1.295596939 + -0.5 Combine like terms: 1.295596939 + -0.5 = 0.795596939 x = 0.795596939 Simplifying x = 0.795596939

Subproblem 2

x + 0.5 = -1.295596939 Simplifying x + 0.5 = -1.295596939 Reorder the terms: 0.5 + x = -1.295596939 Solving 0.5 + x = -1.295596939 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -1.295596939 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.295596939 + -0.5 x = -1.295596939 + -0.5 Combine like terms: -1.295596939 + -0.5 = -1.795596939 x = -1.795596939 Simplifying x = -1.795596939

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.795596939, -1.795596939}

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