4x^2+32x+6+3x^2-7x+10=0

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Solution for 4x^2+32x+6+3x^2-7x+10=0 equation:


Simplifying
4x2 + 32x + 6 + 3x2 + -7x + 10 = 0

Reorder the terms:
6 + 10 + 32x + -7x + 4x2 + 3x2 = 0

Combine like terms: 6 + 10 = 16
16 + 32x + -7x + 4x2 + 3x2 = 0

Combine like terms: 32x + -7x = 25x
16 + 25x + 4x2 + 3x2 = 0

Combine like terms: 4x2 + 3x2 = 7x2
16 + 25x + 7x2 = 0

Solving
16 + 25x + 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'.
2.285714286 + 3.571428571x + x2 = 0

Move the constant term to the right:

Add '-2.285714286' to each side of the equation.
2.285714286 + 3.571428571x + -2.285714286 + x2 = 0 + -2.285714286

Reorder the terms:
2.285714286 + -2.285714286 + 3.571428571x + x2 = 0 + -2.285714286

Combine like terms: 2.285714286 + -2.285714286 = 0.000000000
0.000000000 + 3.571428571x + x2 = 0 + -2.285714286
3.571428571x + x2 = 0 + -2.285714286

Combine like terms: 0 + -2.285714286 = -2.285714286
3.571428571x + x2 = -2.285714286

The x term is 3.571428571x.  Take half its coefficient (1.785714286).
Square it (3.188775511) and add it to both sides.

Add '3.188775511' to each side of the equation.
3.571428571x + 3.188775511 + x2 = -2.285714286 + 3.188775511

Reorder the terms:
3.188775511 + 3.571428571x + x2 = -2.285714286 + 3.188775511

Combine like terms: -2.285714286 + 3.188775511 = 0.903061225
3.188775511 + 3.571428571x + x2 = 0.903061225

Factor a perfect square on the left side:
(x + 1.785714286)(x + 1.785714286) = 0.903061225

Calculate the square root of the right side: 0.950295336

Break this problem into two subproblems by setting 
(x + 1.785714286) equal to 0.950295336 and -0.950295336.

Subproblem 1

x + 1.785714286 = 0.950295336 Simplifying x + 1.785714286 = 0.950295336 Reorder the terms: 1.785714286 + x = 0.950295336 Solving 1.785714286 + x = 0.950295336 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + x = 0.950295336 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + x = 0.950295336 + -1.785714286 x = 0.950295336 + -1.785714286 Combine like terms: 0.950295336 + -1.785714286 = -0.83541895 x = -0.83541895 Simplifying x = -0.83541895

Subproblem 2

x + 1.785714286 = -0.950295336 Simplifying x + 1.785714286 = -0.950295336 Reorder the terms: 1.785714286 + x = -0.950295336 Solving 1.785714286 + x = -0.950295336 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + x = -0.950295336 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + x = -0.950295336 + -1.785714286 x = -0.950295336 + -1.785714286 Combine like terms: -0.950295336 + -1.785714286 = -2.736009622 x = -2.736009622 Simplifying x = -2.736009622

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.83541895, -2.736009622}

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