4x^2-9x-2.5=0

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Solution for 4x^2-9x-2.5=0 equation:


Simplifying
4x2 + -9x + -2.5 = 0

Reorder the terms:
-2.5 + -9x + 4x2 = 0

Solving
-2.5 + -9x + 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'.
-0.625 + -2.25x + x2 = 0

Move the constant term to the right:

Add '0.625' to each side of the equation.
-0.625 + -2.25x + 0.625 + x2 = 0 + 0.625

Reorder the terms:
-0.625 + 0.625 + -2.25x + x2 = 0 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + -2.25x + x2 = 0 + 0.625
-2.25x + x2 = 0 + 0.625

Combine like terms: 0 + 0.625 = 0.625
-2.25x + x2 = 0.625

The x term is -2.25x.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25x + 1.265625 + x2 = 0.625 + 1.265625

Reorder the terms:
1.265625 + -2.25x + x2 = 0.625 + 1.265625

Combine like terms: 0.625 + 1.265625 = 1.890625
1.265625 + -2.25x + x2 = 1.890625

Factor a perfect square on the left side:
(x + -1.125)(x + -1.125) = 1.890625

Calculate the square root of the right side: 1.375

Break this problem into two subproblems by setting 
(x + -1.125) equal to 1.375 and -1.375.

Subproblem 1

x + -1.125 = 1.375 Simplifying x + -1.125 = 1.375 Reorder the terms: -1.125 + x = 1.375 Solving -1.125 + x = 1.375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + x = 1.375 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = 1.375 + 1.125 x = 1.375 + 1.125 Combine like terms: 1.375 + 1.125 = 2.5 x = 2.5 Simplifying x = 2.5

Subproblem 2

x + -1.125 = -1.375 Simplifying x + -1.125 = -1.375 Reorder the terms: -1.125 + x = -1.375 Solving -1.125 + x = -1.375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + x = -1.375 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = -1.375 + 1.125 x = -1.375 + 1.125 Combine like terms: -1.375 + 1.125 = -0.25 x = -0.25 Simplifying x = -0.25

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.5, -0.25}

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