42=.5x(x-5)

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Solution for 42=.5x(x-5) equation:


Simplifying
42 = 0.5x(x + -5)

Reorder the terms:
42 = 0.5x(-5 + x)
42 = (-5 * 0.5x + x * 0.5x)
42 = (-2.5x + 0.5x2)

Solving
42 = -2.5x + 0.5x2

Solving for variable 'x'.

Reorder the terms:
42 + 2.5x + -0.5x2 = -2.5x + 2.5x + 0.5x2 + -0.5x2

Combine like terms: -2.5x + 2.5x = 0.0
42 + 2.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2
42 + 2.5x + -0.5x2 = 0.5x2 + -0.5x2

Combine like terms: 0.5x2 + -0.5x2 = 0.0
42 + 2.5x + -0.5x2 = 0.0

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

Divide each side by '-0.5'.
-84 + -5x + x2 = 0

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + -5x + 84 + x2 = 0 + 84

Reorder the terms:
-84 + 84 + -5x + x2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + -5x + x2 = 0 + 84
-5x + x2 = 0 + 84

Combine like terms: 0 + 84 = 84
-5x + x2 = 84

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = 84 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 84 + 6.25

Combine like terms: 84 + 6.25 = 90.25
6.25 + -5x + x2 = 90.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 90.25

Calculate the square root of the right side: 9.5

Break this problem into two subproblems by setting 
(x + -2.5) equal to 9.5 and -9.5.

Subproblem 1

x + -2.5 = 9.5 Simplifying x + -2.5 = 9.5 Reorder the terms: -2.5 + x = 9.5 Solving -2.5 + x = 9.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 9.5 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 9.5 + 2.5 x = 9.5 + 2.5 Combine like terms: 9.5 + 2.5 = 12 x = 12 Simplifying x = 12

Subproblem 2

x + -2.5 = -9.5 Simplifying x + -2.5 = -9.5 Reorder the terms: -2.5 + x = -9.5 Solving -2.5 + x = -9.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -9.5 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -9.5 + 2.5 x = -9.5 + 2.5 Combine like terms: -9.5 + 2.5 = -7 x = -7 Simplifying x = -7

Solution

The solution to the problem is based on the solutions from the subproblems. x = {12, -7}

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