12x^2-27x-10=0

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Solution for 12x^2-27x-10=0 equation:


Simplifying
12x2 + -27x + -10 = 0

Reorder the terms:
-10 + -27x + 12x2 = 0

Solving
-10 + -27x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
-0.8333333333 + -2.25x + x2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + -2.25x + 0.8333333333 + x2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + -2.25x + x2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + -2.25x + x2 = 0 + 0.8333333333
-2.25x + x2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
-2.25x + x2 = 0.8333333333

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 = 0.8333333333 + 1.265625

Reorder the terms:
1.265625 + -2.25x + x2 = 0.8333333333 + 1.265625

Combine like terms: 0.8333333333 + 1.265625 = 2.0989583333
1.265625 + -2.25x + x2 = 2.0989583333

Factor a perfect square on the left side:
(x + -1.125)(x + -1.125) = 2.0989583333

Calculate the square root of the right side: 1.448778221

Break this problem into two subproblems by setting 
(x + -1.125) equal to 1.448778221 and -1.448778221.

Subproblem 1

x + -1.125 = 1.448778221 Simplifying x + -1.125 = 1.448778221 Reorder the terms: -1.125 + x = 1.448778221 Solving -1.125 + x = 1.448778221 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 = 1.448778221 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = 1.448778221 + 1.125 x = 1.448778221 + 1.125 Combine like terms: 1.448778221 + 1.125 = 2.573778221 x = 2.573778221 Simplifying x = 2.573778221

Subproblem 2

x + -1.125 = -1.448778221 Simplifying x + -1.125 = -1.448778221 Reorder the terms: -1.125 + x = -1.448778221 Solving -1.125 + x = -1.448778221 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 = -1.448778221 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = -1.448778221 + 1.125 x = -1.448778221 + 1.125 Combine like terms: -1.448778221 + 1.125 = -0.323778221 x = -0.323778221 Simplifying x = -0.323778221

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.573778221, -0.323778221}

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