2x(3x-5)-7=20

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Solution for 2x(3x-5)-7=20 equation:


Simplifying
2x(3x + -5) + -7 = 20

Reorder the terms:
2x(-5 + 3x) + -7 = 20
(-5 * 2x + 3x * 2x) + -7 = 20
(-10x + 6x2) + -7 = 20

Reorder the terms:
-7 + -10x + 6x2 = 20

Solving
-7 + -10x + 6x2 = 20

Solving for variable 'x'.

Reorder the terms:
-7 + -20 + -10x + 6x2 = 20 + -20

Combine like terms: -7 + -20 = -27
-27 + -10x + 6x2 = 20 + -20

Combine like terms: 20 + -20 = 0
-27 + -10x + 6x2 = 0

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

Divide each side by '6'.
-4.5 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -1.666666667x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -1.666666667x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -1.666666667x + x2 = 0 + 4.5
-1.666666667x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-1.666666667x + x2 = 4.5

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 4.5 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 4.5 + 0.6944444447

Combine like terms: 4.5 + 0.6944444447 = 5.1944444447
0.6944444447 + -1.666666667x + x2 = 5.1944444447

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 5.1944444447

Calculate the square root of the right side: 2.279132389

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 2.279132389 and -2.279132389.

Subproblem 1

x + -0.8333333335 = 2.279132389 Simplifying x + -0.8333333335 = 2.279132389 Reorder the terms: -0.8333333335 + x = 2.279132389 Solving -0.8333333335 + x = 2.279132389 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 2.279132389 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 2.279132389 + 0.8333333335 x = 2.279132389 + 0.8333333335 Combine like terms: 2.279132389 + 0.8333333335 = 3.1124657225 x = 3.1124657225 Simplifying x = 3.1124657225

Subproblem 2

x + -0.8333333335 = -2.279132389 Simplifying x + -0.8333333335 = -2.279132389 Reorder the terms: -0.8333333335 + x = -2.279132389 Solving -0.8333333335 + x = -2.279132389 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -2.279132389 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -2.279132389 + 0.8333333335 x = -2.279132389 + 0.8333333335 Combine like terms: -2.279132389 + 0.8333333335 = -1.4457990555 x = -1.4457990555 Simplifying x = -1.4457990555

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.1124657225, -1.4457990555}

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