-0.5x^2+10x=x+16

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Solution for -0.5x^2+10x=x+16 equation:


Simplifying
-0.5x2 + 10x = x + 16

Reorder the terms:
10x + -0.5x2 = x + 16

Reorder the terms:
10x + -0.5x2 = 16 + x

Solving
10x + -0.5x2 = 16 + x

Solving for variable 'x'.

Reorder the terms:
-16 + 10x + -1x + -0.5x2 = 16 + x + -16 + -1x

Combine like terms: 10x + -1x = 9x
-16 + 9x + -0.5x2 = 16 + x + -16 + -1x

Reorder the terms:
-16 + 9x + -0.5x2 = 16 + -16 + x + -1x

Combine like terms: 16 + -16 = 0
-16 + 9x + -0.5x2 = 0 + x + -1x
-16 + 9x + -0.5x2 = x + -1x

Combine like terms: x + -1x = 0
-16 + 9x + -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'.
32 + -18x + x2 = 0

Move the constant term to the right:

Add '-32' to each side of the equation.
32 + -18x + -32 + x2 = 0 + -32

Reorder the terms:
32 + -32 + -18x + x2 = 0 + -32

Combine like terms: 32 + -32 = 0
0 + -18x + x2 = 0 + -32
-18x + x2 = 0 + -32

Combine like terms: 0 + -32 = -32
-18x + x2 = -32

The x term is -18x.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18x + 81 + x2 = -32 + 81

Reorder the terms:
81 + -18x + x2 = -32 + 81

Combine like terms: -32 + 81 = 49
81 + -18x + x2 = 49

Factor a perfect square on the left side:
(x + -9)(x + -9) = 49

Calculate the square root of the right side: 7

Break this problem into two subproblems by setting 
(x + -9) equal to 7 and -7.

Subproblem 1

x + -9 = 7 Simplifying x + -9 = 7 Reorder the terms: -9 + x = 7 Solving -9 + x = 7 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x = 7 + 9 Combine like terms: -9 + 9 = 0 0 + x = 7 + 9 x = 7 + 9 Combine like terms: 7 + 9 = 16 x = 16 Simplifying x = 16

Subproblem 2

x + -9 = -7 Simplifying x + -9 = -7 Reorder the terms: -9 + x = -7 Solving -9 + x = -7 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x = -7 + 9 Combine like terms: -9 + 9 = 0 0 + x = -7 + 9 x = -7 + 9 Combine like terms: -7 + 9 = 2 x = 2 Simplifying x = 2

Solution

The solution to the problem is based on the solutions from the subproblems. x = {16, 2}

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