3x^2+2.5x-3=0

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Solution for 3x^2+2.5x-3=0 equation:


Simplifying
3x2 + 2.5x + -3 = 0

Reorder the terms:
-3 + 2.5x + 3x2 = 0

Solving
-3 + 2.5x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-1 + 0.8333333333x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.8333333333x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.8333333333x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.8333333333x + x2 = 0 + 1
0.8333333333x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.8333333333x + x2 = 1

The x term is 0.8333333333x.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333x + 0.1736111111 + x2 = 1 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333x + x2 = 1 + 0.1736111111

Combine like terms: 1 + 0.1736111111 = 1.1736111111
0.1736111111 + 0.8333333333x + x2 = 1.1736111111

Factor a perfect square on the left side:
(x + 0.4166666667)(x + 0.4166666667) = 1.1736111111

Calculate the square root of the right side: 1.083333333

Break this problem into two subproblems by setting 
(x + 0.4166666667) equal to 1.083333333 and -1.083333333.

Subproblem 1

x + 0.4166666667 = 1.083333333 Simplifying x + 0.4166666667 = 1.083333333 Reorder the terms: 0.4166666667 + x = 1.083333333 Solving 0.4166666667 + x = 1.083333333 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = 1.083333333 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 1.083333333 + -0.4166666667 x = 1.083333333 + -0.4166666667 Combine like terms: 1.083333333 + -0.4166666667 = 0.6666666663 x = 0.6666666663 Simplifying x = 0.6666666663

Subproblem 2

x + 0.4166666667 = -1.083333333 Simplifying x + 0.4166666667 = -1.083333333 Reorder the terms: 0.4166666667 + x = -1.083333333 Solving 0.4166666667 + x = -1.083333333 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = -1.083333333 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -1.083333333 + -0.4166666667 x = -1.083333333 + -0.4166666667 Combine like terms: -1.083333333 + -0.4166666667 = -1.4999999997 x = -1.4999999997 Simplifying x = -1.4999999997

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.6666666663, -1.4999999997}

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