4x^2-3x-14=0

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Solution for 4x^2-3x-14=0 equation:


Simplifying
4x2 + -3x + -14 = 0

Reorder the terms:
-14 + -3x + 4x2 = 0

Solving
-14 + -3x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-3.5 + -0.75x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -0.75x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -0.75x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -0.75x + x2 = 0 + 3.5
-0.75x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-0.75x + x2 = 3.5

The x term is -0.75x.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75x + 0.140625 + x2 = 3.5 + 0.140625

Reorder the terms:
0.140625 + -0.75x + x2 = 3.5 + 0.140625

Combine like terms: 3.5 + 0.140625 = 3.640625
0.140625 + -0.75x + x2 = 3.640625

Factor a perfect square on the left side:
(x + -0.375)(x + -0.375) = 3.640625

Calculate the square root of the right side: 1.90804219

Break this problem into two subproblems by setting 
(x + -0.375) equal to 1.90804219 and -1.90804219.

Subproblem 1

x + -0.375 = 1.90804219 Simplifying x + -0.375 = 1.90804219 Reorder the terms: -0.375 + x = 1.90804219 Solving -0.375 + x = 1.90804219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 1.90804219 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 1.90804219 + 0.375 x = 1.90804219 + 0.375 Combine like terms: 1.90804219 + 0.375 = 2.28304219 x = 2.28304219 Simplifying x = 2.28304219

Subproblem 2

x + -0.375 = -1.90804219 Simplifying x + -0.375 = -1.90804219 Reorder the terms: -0.375 + x = -1.90804219 Solving -0.375 + x = -1.90804219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -1.90804219 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -1.90804219 + 0.375 x = -1.90804219 + 0.375 Combine like terms: -1.90804219 + 0.375 = -1.53304219 x = -1.53304219 Simplifying x = -1.53304219

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.28304219, -1.53304219}

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