(x-3)(2x+7)=1

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Solution for (x-3)(2x+7)=1 equation:


Simplifying
(x + -3)(2x + 7) = 1

Reorder the terms:
(-3 + x)(2x + 7) = 1

Reorder the terms:
(-3 + x)(7 + 2x) = 1

Multiply (-3 + x) * (7 + 2x)
(-3(7 + 2x) + x(7 + 2x)) = 1
((7 * -3 + 2x * -3) + x(7 + 2x)) = 1
((-21 + -6x) + x(7 + 2x)) = 1
(-21 + -6x + (7 * x + 2x * x)) = 1
(-21 + -6x + (7x + 2x2)) = 1

Combine like terms: -6x + 7x = 1x
(-21 + 1x + 2x2) = 1

Solving
-21 + 1x + 2x2 = 1

Solving for variable 'x'.

Reorder the terms:
-21 + -1 + 1x + 2x2 = 1 + -1

Combine like terms: -21 + -1 = -22
-22 + 1x + 2x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-22 + 1x + 2x2 = 0

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

Divide each side by '2'.
-11 + 0.5x + x2 = 0

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + 0.5x + 11 + x2 = 0 + 11

Reorder the terms:
-11 + 11 + 0.5x + x2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + 0.5x + x2 = 0 + 11
0.5x + x2 = 0 + 11

Combine like terms: 0 + 11 = 11
0.5x + x2 = 11

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 11 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 11 + 0.0625

Combine like terms: 11 + 0.0625 = 11.0625
0.0625 + 0.5x + x2 = 11.0625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 11.0625

Calculate the square root of the right side: 3.326033674

Break this problem into two subproblems by setting 
(x + 0.25) equal to 3.326033674 and -3.326033674.

Subproblem 1

x + 0.25 = 3.326033674 Simplifying x + 0.25 = 3.326033674 Reorder the terms: 0.25 + x = 3.326033674 Solving 0.25 + x = 3.326033674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 3.326033674 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 3.326033674 + -0.25 x = 3.326033674 + -0.25 Combine like terms: 3.326033674 + -0.25 = 3.076033674 x = 3.076033674 Simplifying x = 3.076033674

Subproblem 2

x + 0.25 = -3.326033674 Simplifying x + 0.25 = -3.326033674 Reorder the terms: 0.25 + x = -3.326033674 Solving 0.25 + x = -3.326033674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -3.326033674 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -3.326033674 + -0.25 x = -3.326033674 + -0.25 Combine like terms: -3.326033674 + -0.25 = -3.576033674 x = -3.576033674 Simplifying x = -3.576033674

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.076033674, -3.576033674}

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