2x^2-3x-75=0

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Solution for 2x^2-3x-75=0 equation:


Simplifying
2x2 + -3x + -75 = 0

Reorder the terms:
-75 + -3x + 2x2 = 0

Solving
-75 + -3x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-37.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '37.5' to each side of the equation.
-37.5 + -1.5x + 37.5 + x2 = 0 + 37.5

Reorder the terms:
-37.5 + 37.5 + -1.5x + x2 = 0 + 37.5

Combine like terms: -37.5 + 37.5 = 0.0
0.0 + -1.5x + x2 = 0 + 37.5
-1.5x + x2 = 0 + 37.5

Combine like terms: 0 + 37.5 = 37.5
-1.5x + x2 = 37.5

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 = 37.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 37.5 + 0.5625

Combine like terms: 37.5 + 0.5625 = 38.0625
0.5625 + -1.5x + x2 = 38.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 38.0625

Calculate the square root of the right side: 6.16948134

Break this problem into two subproblems by setting 
(x + -0.75) equal to 6.16948134 and -6.16948134.

Subproblem 1

x + -0.75 = 6.16948134 Simplifying x + -0.75 = 6.16948134 Reorder the terms: -0.75 + x = 6.16948134 Solving -0.75 + x = 6.16948134 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 = 6.16948134 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 6.16948134 + 0.75 x = 6.16948134 + 0.75 Combine like terms: 6.16948134 + 0.75 = 6.91948134 x = 6.91948134 Simplifying x = 6.91948134

Subproblem 2

x + -0.75 = -6.16948134 Simplifying x + -0.75 = -6.16948134 Reorder the terms: -0.75 + x = -6.16948134 Solving -0.75 + x = -6.16948134 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 = -6.16948134 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -6.16948134 + 0.75 x = -6.16948134 + 0.75 Combine like terms: -6.16948134 + 0.75 = -5.41948134 x = -5.41948134 Simplifying x = -5.41948134

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.91948134, -5.41948134}

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