3.2x^2+4.5x-7.8=0

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Solution for 3.2x^2+4.5x-7.8=0 equation:


Simplifying
3.2x2 + 4.5x + -7.8 = 0

Reorder the terms:
-7.8 + 4.5x + 3.2x2 = 0

Solving
-7.8 + 4.5x + 3.2x2 = 0

Solving for variable 'x'.

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

Divide each side by '3.2'.
-2.4375 + 1.40625x + x2 = 0

Move the constant term to the right:

Add '2.4375' to each side of the equation.
-2.4375 + 1.40625x + 2.4375 + x2 = 0 + 2.4375

Reorder the terms:
-2.4375 + 2.4375 + 1.40625x + x2 = 0 + 2.4375

Combine like terms: -2.4375 + 2.4375 = 0.0000
0.0000 + 1.40625x + x2 = 0 + 2.4375
1.40625x + x2 = 0 + 2.4375

Combine like terms: 0 + 2.4375 = 2.4375
1.40625x + x2 = 2.4375

The x term is 1.40625x.  Take half its coefficient (0.703125).
Square it (0.4943847656) and add it to both sides.

Add '0.4943847656' to each side of the equation.
1.40625x + 0.4943847656 + x2 = 2.4375 + 0.4943847656

Reorder the terms:
0.4943847656 + 1.40625x + x2 = 2.4375 + 0.4943847656

Combine like terms: 2.4375 + 0.4943847656 = 2.9318847656
0.4943847656 + 1.40625x + x2 = 2.9318847656

Factor a perfect square on the left side:
(x + 0.703125)(x + 0.703125) = 2.9318847656

Calculate the square root of the right side: 1.712274734

Break this problem into two subproblems by setting 
(x + 0.703125) equal to 1.712274734 and -1.712274734.

Subproblem 1

x + 0.703125 = 1.712274734 Simplifying x + 0.703125 = 1.712274734 Reorder the terms: 0.703125 + x = 1.712274734 Solving 0.703125 + x = 1.712274734 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.703125' to each side of the equation. 0.703125 + -0.703125 + x = 1.712274734 + -0.703125 Combine like terms: 0.703125 + -0.703125 = 0.000000 0.000000 + x = 1.712274734 + -0.703125 x = 1.712274734 + -0.703125 Combine like terms: 1.712274734 + -0.703125 = 1.009149734 x = 1.009149734 Simplifying x = 1.009149734

Subproblem 2

x + 0.703125 = -1.712274734 Simplifying x + 0.703125 = -1.712274734 Reorder the terms: 0.703125 + x = -1.712274734 Solving 0.703125 + x = -1.712274734 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.703125' to each side of the equation. 0.703125 + -0.703125 + x = -1.712274734 + -0.703125 Combine like terms: 0.703125 + -0.703125 = 0.000000 0.000000 + x = -1.712274734 + -0.703125 x = -1.712274734 + -0.703125 Combine like terms: -1.712274734 + -0.703125 = -2.415399734 x = -2.415399734 Simplifying x = -2.415399734

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.009149734, -2.415399734}

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