4x^2-3x-5=0

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Solution for 4x^2-3x-5=0 equation:


Simplifying
4x2 + -3x + -5 = 0

Reorder the terms:
-5 + -3x + 4x2 = 0

Solving
-5 + -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'.
-1.25 + -0.75x + x2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -0.75x + 1.25 + x2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -0.75x + x2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -0.75x + x2 = 0 + 1.25
-0.75x + x2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-0.75x + x2 = 1.25

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 = 1.25 + 0.140625

Reorder the terms:
0.140625 + -0.75x + x2 = 1.25 + 0.140625

Combine like terms: 1.25 + 0.140625 = 1.390625
0.140625 + -0.75x + x2 = 1.390625

Factor a perfect square on the left side:
(x + -0.375)(x + -0.375) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(x + -0.375) equal to 1.179247642 and -1.179247642.

Subproblem 1

x + -0.375 = 1.179247642 Simplifying x + -0.375 = 1.179247642 Reorder the terms: -0.375 + x = 1.179247642 Solving -0.375 + x = 1.179247642 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 = 1.179247642 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 1.179247642 + 0.375 x = 1.179247642 + 0.375 Combine like terms: 1.179247642 + 0.375 = 1.554247642 x = 1.554247642 Simplifying x = 1.554247642

Subproblem 2

x + -0.375 = -1.179247642 Simplifying x + -0.375 = -1.179247642 Reorder the terms: -0.375 + x = -1.179247642 Solving -0.375 + x = -1.179247642 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 = -1.179247642 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -1.179247642 + 0.375 x = -1.179247642 + 0.375 Combine like terms: -1.179247642 + 0.375 = -0.804247642 x = -0.804247642 Simplifying x = -0.804247642

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.554247642, -0.804247642}

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