0.5(x-2)(x-7)=18

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Solution for 0.5(x-2)(x-7)=18 equation:


Simplifying
0.5(x + -2)(x + -7) = 18

Reorder the terms:
0.5(-2 + x)(x + -7) = 18

Reorder the terms:
0.5(-2 + x)(-7 + x) = 18

Multiply (-2 + x) * (-7 + x)
0.5(-2(-7 + x) + x(-7 + x)) = 18
0.5((-7 * -2 + x * -2) + x(-7 + x)) = 18
0.5((14 + -2x) + x(-7 + x)) = 18
0.5(14 + -2x + (-7 * x + x * x)) = 18
0.5(14 + -2x + (-7x + x2)) = 18

Combine like terms: -2x + -7x = -9x
0.5(14 + -9x + x2) = 18
(14 * 0.5 + -9x * 0.5 + x2 * 0.5) = 18
(7 + -4.5x + 0.5x2) = 18

Solving
7 + -4.5x + 0.5x2 = 18

Solving for variable 'x'.

Reorder the terms:
7 + -18 + -4.5x + 0.5x2 = 18 + -18

Combine like terms: 7 + -18 = -11
-11 + -4.5x + 0.5x2 = 18 + -18

Combine like terms: 18 + -18 = 0
-11 + -4.5x + 0.5x2 = 0

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

Divide each side by '0.5'.
-22 + -9x + x2 = 0

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + -9x + 22 + x2 = 0 + 22

Reorder the terms:
-22 + 22 + -9x + x2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + -9x + x2 = 0 + 22
-9x + x2 = 0 + 22

Combine like terms: 0 + 22 = 22
-9x + x2 = 22

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = 22 + 20.25

Reorder the terms:
20.25 + -9x + x2 = 22 + 20.25

Combine like terms: 22 + 20.25 = 42.25
20.25 + -9x + x2 = 42.25

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 42.25

Calculate the square root of the right side: 6.5

Break this problem into two subproblems by setting 
(x + -4.5) equal to 6.5 and -6.5.

Subproblem 1

x + -4.5 = 6.5 Simplifying x + -4.5 = 6.5 Reorder the terms: -4.5 + x = 6.5 Solving -4.5 + x = 6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = 6.5 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = 6.5 + 4.5 x = 6.5 + 4.5 Combine like terms: 6.5 + 4.5 = 11 x = 11 Simplifying x = 11

Subproblem 2

x + -4.5 = -6.5 Simplifying x + -4.5 = -6.5 Reorder the terms: -4.5 + x = -6.5 Solving -4.5 + x = -6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = -6.5 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = -6.5 + 4.5 x = -6.5 + 4.5 Combine like terms: -6.5 + 4.5 = -2 x = -2 Simplifying x = -2

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11, -2}

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