(a-10)(a-14)=1

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Solution for (a-10)(a-14)=1 equation:


Simplifying
(a + -10)(a + -14) = 1

Reorder the terms:
(-10 + a)(a + -14) = 1

Reorder the terms:
(-10 + a)(-14 + a) = 1

Multiply (-10 + a) * (-14 + a)
(-10(-14 + a) + a(-14 + a)) = 1
((-14 * -10 + a * -10) + a(-14 + a)) = 1
((140 + -10a) + a(-14 + a)) = 1
(140 + -10a + (-14 * a + a * a)) = 1
(140 + -10a + (-14a + a2)) = 1

Combine like terms: -10a + -14a = -24a
(140 + -24a + a2) = 1

Solving
140 + -24a + a2 = 1

Solving for variable 'a'.

Reorder the terms:
140 + -1 + -24a + a2 = 1 + -1

Combine like terms: 140 + -1 = 139
139 + -24a + a2 = 1 + -1

Combine like terms: 1 + -1 = 0
139 + -24a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-139' to each side of the equation.
139 + -24a + -139 + a2 = 0 + -139

Reorder the terms:
139 + -139 + -24a + a2 = 0 + -139

Combine like terms: 139 + -139 = 0
0 + -24a + a2 = 0 + -139
-24a + a2 = 0 + -139

Combine like terms: 0 + -139 = -139
-24a + a2 = -139

The a term is -24a.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24a + 144 + a2 = -139 + 144

Reorder the terms:
144 + -24a + a2 = -139 + 144

Combine like terms: -139 + 144 = 5
144 + -24a + a2 = 5

Factor a perfect square on the left side:
(a + -12)(a + -12) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(a + -12) equal to 2.236067978 and -2.236067978.

Subproblem 1

a + -12 = 2.236067978 Simplifying a + -12 = 2.236067978 Reorder the terms: -12 + a = 2.236067978 Solving -12 + a = 2.236067978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + a = 2.236067978 + 12 Combine like terms: -12 + 12 = 0 0 + a = 2.236067978 + 12 a = 2.236067978 + 12 Combine like terms: 2.236067978 + 12 = 14.236067978 a = 14.236067978 Simplifying a = 14.236067978

Subproblem 2

a + -12 = -2.236067978 Simplifying a + -12 = -2.236067978 Reorder the terms: -12 + a = -2.236067978 Solving -12 + a = -2.236067978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + a = -2.236067978 + 12 Combine like terms: -12 + 12 = 0 0 + a = -2.236067978 + 12 a = -2.236067978 + 12 Combine like terms: -2.236067978 + 12 = 9.763932022 a = 9.763932022 Simplifying a = 9.763932022

Solution

The solution to the problem is based on the solutions from the subproblems. a = {14.236067978, 9.763932022}

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