2x^2+3x=45

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Solution for 2x^2+3x=45 equation:


Simplifying
2x2 + 3x = 45

Reorder the terms:
3x + 2x2 = 45

Solving
3x + 2x2 = 45

Solving for variable 'x'.

Reorder the terms:
-45 + 3x + 2x2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 3x + 2x2 = 0

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

Divide each side by '2'.
-22.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + 1.5x + 22.5 + x2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + 1.5x + x2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + 1.5x + x2 = 0 + 22.5
1.5x + x2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
1.5x + x2 = 22.5

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 = 22.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 22.5 + 0.5625

Combine like terms: 22.5 + 0.5625 = 23.0625
0.5625 + 1.5x + x2 = 23.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 23.0625

Calculate the square root of the right side: 4.802343178

Break this problem into two subproblems by setting 
(x + 0.75) equal to 4.802343178 and -4.802343178.

Subproblem 1

x + 0.75 = 4.802343178 Simplifying x + 0.75 = 4.802343178 Reorder the terms: 0.75 + x = 4.802343178 Solving 0.75 + x = 4.802343178 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 = 4.802343178 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 4.802343178 + -0.75 x = 4.802343178 + -0.75 Combine like terms: 4.802343178 + -0.75 = 4.052343178 x = 4.052343178 Simplifying x = 4.052343178

Subproblem 2

x + 0.75 = -4.802343178 Simplifying x + 0.75 = -4.802343178 Reorder the terms: 0.75 + x = -4.802343178 Solving 0.75 + x = -4.802343178 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 = -4.802343178 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -4.802343178 + -0.75 x = -4.802343178 + -0.75 Combine like terms: -4.802343178 + -0.75 = -5.552343178 x = -5.552343178 Simplifying x = -5.552343178

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.052343178, -5.552343178}

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