2x^2+3x=105

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Solution for 2x^2+3x=105 equation:


Simplifying
2x2 + 3x = 105

Reorder the terms:
3x + 2x2 = 105

Solving
3x + 2x2 = 105

Solving for variable 'x'.

Reorder the terms:
-105 + 3x + 2x2 = 105 + -105

Combine like terms: 105 + -105 = 0
-105 + 3x + 2x2 = 0

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

Divide each side by '2'.
-52.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '52.5' to each side of the equation.
-52.5 + 1.5x + 52.5 + x2 = 0 + 52.5

Reorder the terms:
-52.5 + 52.5 + 1.5x + x2 = 0 + 52.5

Combine like terms: -52.5 + 52.5 = 0.0
0.0 + 1.5x + x2 = 0 + 52.5
1.5x + x2 = 0 + 52.5

Combine like terms: 0 + 52.5 = 52.5
1.5x + x2 = 52.5

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 52.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 52.5 + 0.5625

Combine like terms: 52.5 + 0.5625 = 53.0625
0.5625 + 1.5x + x2 = 53.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 53.0625

Calculate the square root of the right side: 7.284401142

Break this problem into two subproblems by setting 
(x + 0.75) equal to 7.284401142 and -7.284401142.

Subproblem 1

x + 0.75 = 7.284401142 Simplifying x + 0.75 = 7.284401142 Reorder the terms: 0.75 + x = 7.284401142 Solving 0.75 + x = 7.284401142 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 7.284401142 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 7.284401142 + -0.75 x = 7.284401142 + -0.75 Combine like terms: 7.284401142 + -0.75 = 6.534401142 x = 6.534401142 Simplifying x = 6.534401142

Subproblem 2

x + 0.75 = -7.284401142 Simplifying x + 0.75 = -7.284401142 Reorder the terms: 0.75 + x = -7.284401142 Solving 0.75 + x = -7.284401142 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -7.284401142 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -7.284401142 + -0.75 x = -7.284401142 + -0.75 Combine like terms: -7.284401142 + -0.75 = -8.034401142 x = -8.034401142 Simplifying x = -8.034401142

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.534401142, -8.034401142}

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