2x^2=19-3x

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Solution for 2x^2=19-3x equation:


Simplifying
2x2 = 19 + -3x

Solving
2x2 = 19 + -3x

Solving for variable 'x'.

Reorder the terms:
-19 + 3x + 2x2 = 19 + -3x + -19 + 3x

Reorder the terms:
-19 + 3x + 2x2 = 19 + -19 + -3x + 3x

Combine like terms: 19 + -19 = 0
-19 + 3x + 2x2 = 0 + -3x + 3x
-19 + 3x + 2x2 = -3x + 3x

Combine like terms: -3x + 3x = 0
-19 + 3x + 2x2 = 0

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

Divide each side by '2'.
-9.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '9.5' to each side of the equation.
-9.5 + 1.5x + 9.5 + x2 = 0 + 9.5

Reorder the terms:
-9.5 + 9.5 + 1.5x + x2 = 0 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + 1.5x + x2 = 0 + 9.5
1.5x + x2 = 0 + 9.5

Combine like terms: 0 + 9.5 = 9.5
1.5x + x2 = 9.5

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 = 9.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 9.5 + 0.5625

Combine like terms: 9.5 + 0.5625 = 10.0625
0.5625 + 1.5x + x2 = 10.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 10.0625

Calculate the square root of the right side: 3.172144385

Break this problem into two subproblems by setting 
(x + 0.75) equal to 3.172144385 and -3.172144385.

Subproblem 1

x + 0.75 = 3.172144385 Simplifying x + 0.75 = 3.172144385 Reorder the terms: 0.75 + x = 3.172144385 Solving 0.75 + x = 3.172144385 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 = 3.172144385 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 3.172144385 + -0.75 x = 3.172144385 + -0.75 Combine like terms: 3.172144385 + -0.75 = 2.422144385 x = 2.422144385 Simplifying x = 2.422144385

Subproblem 2

x + 0.75 = -3.172144385 Simplifying x + 0.75 = -3.172144385 Reorder the terms: 0.75 + x = -3.172144385 Solving 0.75 + x = -3.172144385 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 = -3.172144385 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -3.172144385 + -0.75 x = -3.172144385 + -0.75 Combine like terms: -3.172144385 + -0.75 = -3.922144385 x = -3.922144385 Simplifying x = -3.922144385

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.422144385, -3.922144385}

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