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

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


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

Reorder the terms:
(-2 + 3x)(5x + -7) = x

Reorder the terms:
(-2 + 3x)(-7 + 5x) = x

Multiply (-2 + 3x) * (-7 + 5x)
(-2(-7 + 5x) + 3x * (-7 + 5x)) = x
((-7 * -2 + 5x * -2) + 3x * (-7 + 5x)) = x
((14 + -10x) + 3x * (-7 + 5x)) = x
(14 + -10x + (-7 * 3x + 5x * 3x)) = x
(14 + -10x + (-21x + 15x2)) = x

Combine like terms: -10x + -21x = -31x
(14 + -31x + 15x2) = x

Solving
14 + -31x + 15x2 = x

Solving for variable 'x'.

Reorder the terms:
14 + -31x + -1x + 15x2 = x + -1x

Combine like terms: -31x + -1x = -32x
14 + -32x + 15x2 = x + -1x

Combine like terms: x + -1x = 0
14 + -32x + 15x2 = 0

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

Divide each side by '15'.
0.9333333333 + -2.133333333x + x2 = 0

Move the constant term to the right:

Add '-0.9333333333' to each side of the equation.
0.9333333333 + -2.133333333x + -0.9333333333 + x2 = 0 + -0.9333333333

Reorder the terms:
0.9333333333 + -0.9333333333 + -2.133333333x + x2 = 0 + -0.9333333333

Combine like terms: 0.9333333333 + -0.9333333333 = 0.0000000000
0.0000000000 + -2.133333333x + x2 = 0 + -0.9333333333
-2.133333333x + x2 = 0 + -0.9333333333

Combine like terms: 0 + -0.9333333333 = -0.9333333333
-2.133333333x + x2 = -0.9333333333

The x term is -2.133333333x.  Take half its coefficient (-1.066666667).
Square it (1.137777778) and add it to both sides.

Add '1.137777778' to each side of the equation.
-2.133333333x + 1.137777778 + x2 = -0.9333333333 + 1.137777778

Reorder the terms:
1.137777778 + -2.133333333x + x2 = -0.9333333333 + 1.137777778

Combine like terms: -0.9333333333 + 1.137777778 = 0.2044444447
1.137777778 + -2.133333333x + x2 = 0.2044444447

Factor a perfect square on the left side:
(x + -1.066666667)(x + -1.066666667) = 0.2044444447

Calculate the square root of the right side: 0.452155332

Break this problem into two subproblems by setting 
(x + -1.066666667) equal to 0.452155332 and -0.452155332.

Subproblem 1

x + -1.066666667 = 0.452155332 Simplifying x + -1.066666667 = 0.452155332 Reorder the terms: -1.066666667 + x = 0.452155332 Solving -1.066666667 + x = 0.452155332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.066666667' to each side of the equation. -1.066666667 + 1.066666667 + x = 0.452155332 + 1.066666667 Combine like terms: -1.066666667 + 1.066666667 = 0.000000000 0.000000000 + x = 0.452155332 + 1.066666667 x = 0.452155332 + 1.066666667 Combine like terms: 0.452155332 + 1.066666667 = 1.518821999 x = 1.518821999 Simplifying x = 1.518821999

Subproblem 2

x + -1.066666667 = -0.452155332 Simplifying x + -1.066666667 = -0.452155332 Reorder the terms: -1.066666667 + x = -0.452155332 Solving -1.066666667 + x = -0.452155332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.066666667' to each side of the equation. -1.066666667 + 1.066666667 + x = -0.452155332 + 1.066666667 Combine like terms: -1.066666667 + 1.066666667 = 0.000000000 0.000000000 + x = -0.452155332 + 1.066666667 x = -0.452155332 + 1.066666667 Combine like terms: -0.452155332 + 1.066666667 = 0.614511335 x = 0.614511335 Simplifying x = 0.614511335

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.518821999, 0.614511335}

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