42x^2=27x+40

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Solution for 42x^2=27x+40 equation:


Simplifying
42x2 = 27x + 40

Reorder the terms:
42x2 = 40 + 27x

Solving
42x2 = 40 + 27x

Solving for variable 'x'.

Reorder the terms:
-40 + -27x + 42x2 = 40 + 27x + -40 + -27x

Reorder the terms:
-40 + -27x + 42x2 = 40 + -40 + 27x + -27x

Combine like terms: 40 + -40 = 0
-40 + -27x + 42x2 = 0 + 27x + -27x
-40 + -27x + 42x2 = 27x + -27x

Combine like terms: 27x + -27x = 0
-40 + -27x + 42x2 = 0

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

Divide each side by '42'.
-0.9523809524 + -0.6428571429x + x2 = 0

Move the constant term to the right:

Add '0.9523809524' to each side of the equation.
-0.9523809524 + -0.6428571429x + 0.9523809524 + x2 = 0 + 0.9523809524

Reorder the terms:
-0.9523809524 + 0.9523809524 + -0.6428571429x + x2 = 0 + 0.9523809524

Combine like terms: -0.9523809524 + 0.9523809524 = 0.0000000000
0.0000000000 + -0.6428571429x + x2 = 0 + 0.9523809524
-0.6428571429x + x2 = 0 + 0.9523809524

Combine like terms: 0 + 0.9523809524 = 0.9523809524
-0.6428571429x + x2 = 0.9523809524

The x term is -0.6428571429x.  Take half its coefficient (-0.3214285715).
Square it (0.1033163266) and add it to both sides.

Add '0.1033163266' to each side of the equation.
-0.6428571429x + 0.1033163266 + x2 = 0.9523809524 + 0.1033163266

Reorder the terms:
0.1033163266 + -0.6428571429x + x2 = 0.9523809524 + 0.1033163266

Combine like terms: 0.9523809524 + 0.1033163266 = 1.055697279
0.1033163266 + -0.6428571429x + x2 = 1.055697279

Factor a perfect square on the left side:
(x + -0.3214285715)(x + -0.3214285715) = 1.055697279

Calculate the square root of the right side: 1.027471303

Break this problem into two subproblems by setting 
(x + -0.3214285715) equal to 1.027471303 and -1.027471303.

Subproblem 1

x + -0.3214285715 = 1.027471303 Simplifying x + -0.3214285715 = 1.027471303 Reorder the terms: -0.3214285715 + x = 1.027471303 Solving -0.3214285715 + x = 1.027471303 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3214285715' to each side of the equation. -0.3214285715 + 0.3214285715 + x = 1.027471303 + 0.3214285715 Combine like terms: -0.3214285715 + 0.3214285715 = 0.0000000000 0.0000000000 + x = 1.027471303 + 0.3214285715 x = 1.027471303 + 0.3214285715 Combine like terms: 1.027471303 + 0.3214285715 = 1.3488998745 x = 1.3488998745 Simplifying x = 1.3488998745

Subproblem 2

x + -0.3214285715 = -1.027471303 Simplifying x + -0.3214285715 = -1.027471303 Reorder the terms: -0.3214285715 + x = -1.027471303 Solving -0.3214285715 + x = -1.027471303 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3214285715' to each side of the equation. -0.3214285715 + 0.3214285715 + x = -1.027471303 + 0.3214285715 Combine like terms: -0.3214285715 + 0.3214285715 = 0.0000000000 0.0000000000 + x = -1.027471303 + 0.3214285715 x = -1.027471303 + 0.3214285715 Combine like terms: -1.027471303 + 0.3214285715 = -0.7060427315 x = -0.7060427315 Simplifying x = -0.7060427315

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3488998745, -0.7060427315}

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