2x^2+3x+4=45

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Solution for 2x^2+3x+4=45 equation:


Simplifying
2x2 + 3x + 4 = 45

Reorder the terms:
4 + 3x + 2x2 = 45

Solving
4 + 3x + 2x2 = 45

Solving for variable 'x'.

Reorder the terms:
4 + -45 + 3x + 2x2 = 45 + -45

Combine like terms: 4 + -45 = -41
-41 + 3x + 2x2 = 45 + -45

Combine like terms: 45 + -45 = 0
-41 + 3x + 2x2 = 0

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

Divide each side by '2'.
-20.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '20.5' to each side of the equation.
-20.5 + 1.5x + 20.5 + x2 = 0 + 20.5

Reorder the terms:
-20.5 + 20.5 + 1.5x + x2 = 0 + 20.5

Combine like terms: -20.5 + 20.5 = 0.0
0.0 + 1.5x + x2 = 0 + 20.5
1.5x + x2 = 0 + 20.5

Combine like terms: 0 + 20.5 = 20.5
1.5x + x2 = 20.5

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 = 20.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 20.5 + 0.5625

Combine like terms: 20.5 + 0.5625 = 21.0625
0.5625 + 1.5x + x2 = 21.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 21.0625

Calculate the square root of the right side: 4.589389938

Break this problem into two subproblems by setting 
(x + 0.75) equal to 4.589389938 and -4.589389938.

Subproblem 1

x + 0.75 = 4.589389938 Simplifying x + 0.75 = 4.589389938 Reorder the terms: 0.75 + x = 4.589389938 Solving 0.75 + x = 4.589389938 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 = 4.589389938 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 4.589389938 + -0.75 x = 4.589389938 + -0.75 Combine like terms: 4.589389938 + -0.75 = 3.839389938 x = 3.839389938 Simplifying x = 3.839389938

Subproblem 2

x + 0.75 = -4.589389938 Simplifying x + 0.75 = -4.589389938 Reorder the terms: 0.75 + x = -4.589389938 Solving 0.75 + x = -4.589389938 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 = -4.589389938 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -4.589389938 + -0.75 x = -4.589389938 + -0.75 Combine like terms: -4.589389938 + -0.75 = -5.339389938 x = -5.339389938 Simplifying x = -5.339389938

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.839389938, -5.339389938}

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