(3x-4)*(5x+3)=1

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Solution for (3x-4)*(5x+3)=1 equation:


Simplifying
(3x + -4)(5x + 3) = 1

Reorder the terms:
(-4 + 3x)(5x + 3) = 1

Reorder the terms:
(-4 + 3x)(3 + 5x) = 1

Multiply (-4 + 3x) * (3 + 5x)
(-4(3 + 5x) + 3x * (3 + 5x)) = 1
((3 * -4 + 5x * -4) + 3x * (3 + 5x)) = 1
((-12 + -20x) + 3x * (3 + 5x)) = 1
(-12 + -20x + (3 * 3x + 5x * 3x)) = 1
(-12 + -20x + (9x + 15x2)) = 1

Combine like terms: -20x + 9x = -11x
(-12 + -11x + 15x2) = 1

Solving
-12 + -11x + 15x2 = 1

Solving for variable 'x'.

Reorder the terms:
-12 + -1 + -11x + 15x2 = 1 + -1

Combine like terms: -12 + -1 = -13
-13 + -11x + 15x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-13 + -11x + 15x2 = 0

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

Divide each side by '15'.
-0.8666666667 + -0.7333333333x + x2 = 0

Move the constant term to the right:

Add '0.8666666667' to each side of the equation.
-0.8666666667 + -0.7333333333x + 0.8666666667 + x2 = 0 + 0.8666666667

Reorder the terms:
-0.8666666667 + 0.8666666667 + -0.7333333333x + x2 = 0 + 0.8666666667

Combine like terms: -0.8666666667 + 0.8666666667 = 0.0000000000
0.0000000000 + -0.7333333333x + x2 = 0 + 0.8666666667
-0.7333333333x + x2 = 0 + 0.8666666667

Combine like terms: 0 + 0.8666666667 = 0.8666666667
-0.7333333333x + x2 = 0.8666666667

The x term is -0.7333333333x.  Take half its coefficient (-0.3666666667).
Square it (0.1344444445) and add it to both sides.

Add '0.1344444445' to each side of the equation.
-0.7333333333x + 0.1344444445 + x2 = 0.8666666667 + 0.1344444445

Reorder the terms:
0.1344444445 + -0.7333333333x + x2 = 0.8666666667 + 0.1344444445

Combine like terms: 0.8666666667 + 0.1344444445 = 1.0011111112
0.1344444445 + -0.7333333333x + x2 = 1.0011111112

Factor a perfect square on the left side:
(x + -0.3666666667)(x + -0.3666666667) = 1.0011111112

Calculate the square root of the right side: 1.000555401

Break this problem into two subproblems by setting 
(x + -0.3666666667) equal to 1.000555401 and -1.000555401.

Subproblem 1

x + -0.3666666667 = 1.000555401 Simplifying x + -0.3666666667 = 1.000555401 Reorder the terms: -0.3666666667 + x = 1.000555401 Solving -0.3666666667 + x = 1.000555401 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3666666667' to each side of the equation. -0.3666666667 + 0.3666666667 + x = 1.000555401 + 0.3666666667 Combine like terms: -0.3666666667 + 0.3666666667 = 0.0000000000 0.0000000000 + x = 1.000555401 + 0.3666666667 x = 1.000555401 + 0.3666666667 Combine like terms: 1.000555401 + 0.3666666667 = 1.3672220677 x = 1.3672220677 Simplifying x = 1.3672220677

Subproblem 2

x + -0.3666666667 = -1.000555401 Simplifying x + -0.3666666667 = -1.000555401 Reorder the terms: -0.3666666667 + x = -1.000555401 Solving -0.3666666667 + x = -1.000555401 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3666666667' to each side of the equation. -0.3666666667 + 0.3666666667 + x = -1.000555401 + 0.3666666667 Combine like terms: -0.3666666667 + 0.3666666667 = 0.0000000000 0.0000000000 + x = -1.000555401 + 0.3666666667 x = -1.000555401 + 0.3666666667 Combine like terms: -1.000555401 + 0.3666666667 = -0.6338887343 x = -0.6338887343 Simplifying x = -0.6338887343

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3672220677, -0.6338887343}

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