(h-3)(h-5)=1

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Solution for (h-3)(h-5)=1 equation:


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

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

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

Multiply (-3 + h) * (-5 + h)
(-3(-5 + h) + h(-5 + h)) = 1
((-5 * -3 + h * -3) + h(-5 + h)) = 1
((15 + -3h) + h(-5 + h)) = 1
(15 + -3h + (-5 * h + h * h)) = 1
(15 + -3h + (-5h + h2)) = 1

Combine like terms: -3h + -5h = -8h
(15 + -8h + h2) = 1

Solving
15 + -8h + h2 = 1

Solving for variable 'h'.

Reorder the terms:
15 + -1 + -8h + h2 = 1 + -1

Combine like terms: 15 + -1 = 14
14 + -8h + h2 = 1 + -1

Combine like terms: 1 + -1 = 0
14 + -8h + h2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-14' to each side of the equation.
14 + -8h + -14 + h2 = 0 + -14

Reorder the terms:
14 + -14 + -8h + h2 = 0 + -14

Combine like terms: 14 + -14 = 0
0 + -8h + h2 = 0 + -14
-8h + h2 = 0 + -14

Combine like terms: 0 + -14 = -14
-8h + h2 = -14

The h term is -8h.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8h + 16 + h2 = -14 + 16

Reorder the terms:
16 + -8h + h2 = -14 + 16

Combine like terms: -14 + 16 = 2
16 + -8h + h2 = 2

Factor a perfect square on the left side:
(h + -4)(h + -4) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(h + -4) equal to 1.414213562 and -1.414213562.

Subproblem 1

h + -4 = 1.414213562 Simplifying h + -4 = 1.414213562 Reorder the terms: -4 + h = 1.414213562 Solving -4 + h = 1.414213562 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + h = 1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + h = 1.414213562 + 4 h = 1.414213562 + 4 Combine like terms: 1.414213562 + 4 = 5.414213562 h = 5.414213562 Simplifying h = 5.414213562

Subproblem 2

h + -4 = -1.414213562 Simplifying h + -4 = -1.414213562 Reorder the terms: -4 + h = -1.414213562 Solving -4 + h = -1.414213562 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + h = -1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + h = -1.414213562 + 4 h = -1.414213562 + 4 Combine like terms: -1.414213562 + 4 = 2.585786438 h = 2.585786438 Simplifying h = 2.585786438

Solution

The solution to the problem is based on the solutions from the subproblems. h = {5.414213562, 2.585786438}

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