(x)=x^2-14x+45

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Solution for (x)=x^2-14x+45 equation:


Simplifying
(x) = x2 + -14x + 45
x = x2 + -14x + 45

Reorder the terms:
x = 45 + -14x + x2

Solving
x = 45 + -14x + x2

Solving for variable 'x'.

Reorder the terms:
-45 + x + 14x + -1x2 = 45 + -14x + x2 + -45 + 14x + -1x2

Combine like terms: x + 14x = 15x
-45 + 15x + -1x2 = 45 + -14x + x2 + -45 + 14x + -1x2

Reorder the terms:
-45 + 15x + -1x2 = 45 + -45 + -14x + 14x + x2 + -1x2

Combine like terms: 45 + -45 = 0
-45 + 15x + -1x2 = 0 + -14x + 14x + x2 + -1x2
-45 + 15x + -1x2 = -14x + 14x + x2 + -1x2

Combine like terms: -14x + 14x = 0
-45 + 15x + -1x2 = 0 + x2 + -1x2
-45 + 15x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-45 + 15x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
45 + -15x + x2 = 0

Move the constant term to the right:

Add '-45' to each side of the equation.
45 + -15x + -45 + x2 = 0 + -45

Reorder the terms:
45 + -45 + -15x + x2 = 0 + -45

Combine like terms: 45 + -45 = 0
0 + -15x + x2 = 0 + -45
-15x + x2 = 0 + -45

Combine like terms: 0 + -45 = -45
-15x + x2 = -45

The x term is -15x.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15x + 56.25 + x2 = -45 + 56.25

Reorder the terms:
56.25 + -15x + x2 = -45 + 56.25

Combine like terms: -45 + 56.25 = 11.25
56.25 + -15x + x2 = 11.25

Factor a perfect square on the left side:
(x + -7.5)(x + -7.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(x + -7.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

x + -7.5 = 3.354101966 Simplifying x + -7.5 = 3.354101966 Reorder the terms: -7.5 + x = 3.354101966 Solving -7.5 + x = 3.354101966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = 3.354101966 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = 3.354101966 + 7.5 x = 3.354101966 + 7.5 Combine like terms: 3.354101966 + 7.5 = 10.854101966 x = 10.854101966 Simplifying x = 10.854101966

Subproblem 2

x + -7.5 = -3.354101966 Simplifying x + -7.5 = -3.354101966 Reorder the terms: -7.5 + x = -3.354101966 Solving -7.5 + x = -3.354101966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = -3.354101966 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = -3.354101966 + 7.5 x = -3.354101966 + 7.5 Combine like terms: -3.354101966 + 7.5 = 4.145898034 x = 4.145898034 Simplifying x = 4.145898034

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.854101966, 4.145898034}

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