3x^2+10x+3=19-4x^2+16

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Solution for 3x^2+10x+3=19-4x^2+16 equation:


Simplifying
3x2 + 10x + 3 = 19 + -4x2 + 16

Reorder the terms:
3 + 10x + 3x2 = 19 + -4x2 + 16

Reorder the terms:
3 + 10x + 3x2 = 19 + 16 + -4x2

Combine like terms: 19 + 16 = 35
3 + 10x + 3x2 = 35 + -4x2

Solving
3 + 10x + 3x2 = 35 + -4x2

Solving for variable 'x'.

Reorder the terms:
3 + -35 + 10x + 3x2 + 4x2 = 35 + -4x2 + -35 + 4x2

Combine like terms: 3 + -35 = -32
-32 + 10x + 3x2 + 4x2 = 35 + -4x2 + -35 + 4x2

Combine like terms: 3x2 + 4x2 = 7x2
-32 + 10x + 7x2 = 35 + -4x2 + -35 + 4x2

Reorder the terms:
-32 + 10x + 7x2 = 35 + -35 + -4x2 + 4x2

Combine like terms: 35 + -35 = 0
-32 + 10x + 7x2 = 0 + -4x2 + 4x2
-32 + 10x + 7x2 = -4x2 + 4x2

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

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

Divide each side by '7'.
-4.571428571 + 1.428571429x + x2 = 0

Move the constant term to the right:

Add '4.571428571' to each side of the equation.
-4.571428571 + 1.428571429x + 4.571428571 + x2 = 0 + 4.571428571

Reorder the terms:
-4.571428571 + 4.571428571 + 1.428571429x + x2 = 0 + 4.571428571

Combine like terms: -4.571428571 + 4.571428571 = 0.000000000
0.000000000 + 1.428571429x + x2 = 0 + 4.571428571
1.428571429x + x2 = 0 + 4.571428571

Combine like terms: 0 + 4.571428571 = 4.571428571
1.428571429x + x2 = 4.571428571

The x term is 1.428571429x.  Take half its coefficient (0.7142857145).
Square it (0.5102040819) and add it to both sides.

Add '0.5102040819' to each side of the equation.
1.428571429x + 0.5102040819 + x2 = 4.571428571 + 0.5102040819

Reorder the terms:
0.5102040819 + 1.428571429x + x2 = 4.571428571 + 0.5102040819

Combine like terms: 4.571428571 + 0.5102040819 = 5.0816326529
0.5102040819 + 1.428571429x + x2 = 5.0816326529

Factor a perfect square on the left side:
(x + 0.7142857145)(x + 0.7142857145) = 5.0816326529

Calculate the square root of the right side: 2.254247691

Break this problem into two subproblems by setting 
(x + 0.7142857145) equal to 2.254247691 and -2.254247691.

Subproblem 1

x + 0.7142857145 = 2.254247691 Simplifying x + 0.7142857145 = 2.254247691 Reorder the terms: 0.7142857145 + x = 2.254247691 Solving 0.7142857145 + x = 2.254247691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7142857145' to each side of the equation. 0.7142857145 + -0.7142857145 + x = 2.254247691 + -0.7142857145 Combine like terms: 0.7142857145 + -0.7142857145 = 0.0000000000 0.0000000000 + x = 2.254247691 + -0.7142857145 x = 2.254247691 + -0.7142857145 Combine like terms: 2.254247691 + -0.7142857145 = 1.5399619765 x = 1.5399619765 Simplifying x = 1.5399619765

Subproblem 2

x + 0.7142857145 = -2.254247691 Simplifying x + 0.7142857145 = -2.254247691 Reorder the terms: 0.7142857145 + x = -2.254247691 Solving 0.7142857145 + x = -2.254247691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7142857145' to each side of the equation. 0.7142857145 + -0.7142857145 + x = -2.254247691 + -0.7142857145 Combine like terms: 0.7142857145 + -0.7142857145 = 0.0000000000 0.0000000000 + x = -2.254247691 + -0.7142857145 x = -2.254247691 + -0.7142857145 Combine like terms: -2.254247691 + -0.7142857145 = -2.9685334055 x = -2.9685334055 Simplifying x = -2.9685334055

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.5399619765, -2.9685334055}

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