0.5x(x-3)+(1.5-x)=5x

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Solution for 0.5x(x-3)+(1.5-x)=5x equation:


Simplifying
0.5x(x + -3) + (1.5 + -1x) = 5x

Reorder the terms:
0.5x(-3 + x) + (1.5 + -1x) = 5x
(-3 * 0.5x + x * 0.5x) + (1.5 + -1x) = 5x
(-1.5x + 0.5x2) + (1.5 + -1x) = 5x

Remove parenthesis around (1.5 + -1x)
-1.5x + 0.5x2 + 1.5 + -1x = 5x

Reorder the terms:
1.5 + -1.5x + -1x + 0.5x2 = 5x

Combine like terms: -1.5x + -1x = -2.5x
1.5 + -2.5x + 0.5x2 = 5x

Solving
1.5 + -2.5x + 0.5x2 = 5x

Solving for variable 'x'.

Reorder the terms:
1.5 + -2.5x + -5x + 0.5x2 = 5x + -5x

Combine like terms: -2.5x + -5x = -7.5x
1.5 + -7.5x + 0.5x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
1.5 + -7.5x + 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'.
3 + -15x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -15x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -15x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -15x + x2 = 0 + -3
-15x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-15x + x2 = -3

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 = -3 + 56.25

Reorder the terms:
56.25 + -15x + x2 = -3 + 56.25

Combine like terms: -3 + 56.25 = 53.25
56.25 + -15x + x2 = 53.25

Factor a perfect square on the left side:
(x + -7.5)(x + -7.5) = 53.25

Calculate the square root of the right side: 7.29725976

Break this problem into two subproblems by setting 
(x + -7.5) equal to 7.29725976 and -7.29725976.

Subproblem 1

x + -7.5 = 7.29725976 Simplifying x + -7.5 = 7.29725976 Reorder the terms: -7.5 + x = 7.29725976 Solving -7.5 + x = 7.29725976 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 = 7.29725976 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = 7.29725976 + 7.5 x = 7.29725976 + 7.5 Combine like terms: 7.29725976 + 7.5 = 14.79725976 x = 14.79725976 Simplifying x = 14.79725976

Subproblem 2

x + -7.5 = -7.29725976 Simplifying x + -7.5 = -7.29725976 Reorder the terms: -7.5 + x = -7.29725976 Solving -7.5 + x = -7.29725976 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 = -7.29725976 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = -7.29725976 + 7.5 x = -7.29725976 + 7.5 Combine like terms: -7.29725976 + 7.5 = 0.20274024 x = 0.20274024 Simplifying x = 0.20274024

Solution

The solution to the problem is based on the solutions from the subproblems. x = {14.79725976, 0.20274024}

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