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

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


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

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

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

Multiply (-3 + x) * (5 + x)
(-3(5 + x) + x(5 + x)) = 1
((5 * -3 + x * -3) + x(5 + x)) = 1
((-15 + -3x) + x(5 + x)) = 1
(-15 + -3x + (5 * x + x * x)) = 1
(-15 + -3x + (5x + x2)) = 1

Combine like terms: -3x + 5x = 2x
(-15 + 2x + x2) = 1

Solving
-15 + 2x + x2 = 1

Solving for variable 'x'.

Reorder the terms:
-15 + -1 + 2x + x2 = 1 + -1

Combine like terms: -15 + -1 = -16
-16 + 2x + x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-16 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + 2x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + 2x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + 2x + x2 = 0 + 16
2x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
2x + x2 = 16

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 16 + 1

Reorder the terms:
1 + 2x + x2 = 16 + 1

Combine like terms: 16 + 1 = 17
1 + 2x + x2 = 17

Factor a perfect square on the left side:
(x + 1)(x + 1) = 17

Calculate the square root of the right side: 4.123105626

Break this problem into two subproblems by setting 
(x + 1) equal to 4.123105626 and -4.123105626.

Subproblem 1

x + 1 = 4.123105626 Simplifying x + 1 = 4.123105626 Reorder the terms: 1 + x = 4.123105626 Solving 1 + x = 4.123105626 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 4.123105626 + -1 Combine like terms: 1 + -1 = 0 0 + x = 4.123105626 + -1 x = 4.123105626 + -1 Combine like terms: 4.123105626 + -1 = 3.123105626 x = 3.123105626 Simplifying x = 3.123105626

Subproblem 2

x + 1 = -4.123105626 Simplifying x + 1 = -4.123105626 Reorder the terms: 1 + x = -4.123105626 Solving 1 + x = -4.123105626 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -4.123105626 + -1 Combine like terms: 1 + -1 = 0 0 + x = -4.123105626 + -1 x = -4.123105626 + -1 Combine like terms: -4.123105626 + -1 = -5.123105626 x = -5.123105626 Simplifying x = -5.123105626

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.123105626, -5.123105626}

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