(0.5x+1)(x+1)=10

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Solution for (0.5x+1)(x+1)=10 equation:


Simplifying
(0.5x + 1)(x + 1) = 10

Reorder the terms:
(1 + 0.5x)(x + 1) = 10

Reorder the terms:
(1 + 0.5x)(1 + x) = 10

Multiply (1 + 0.5x) * (1 + x)
(1(1 + x) + 0.5x * (1 + x)) = 10
((1 * 1 + x * 1) + 0.5x * (1 + x)) = 10
((1 + 1x) + 0.5x * (1 + x)) = 10
(1 + 1x + (1 * 0.5x + x * 0.5x)) = 10
(1 + 1x + (0.5x + 0.5x2)) = 10

Combine like terms: 1x + 0.5x = 1.5x
(1 + 1.5x + 0.5x2) = 10

Solving
1 + 1.5x + 0.5x2 = 10

Solving for variable 'x'.

Reorder the terms:
1 + -10 + 1.5x + 0.5x2 = 10 + -10

Combine like terms: 1 + -10 = -9
-9 + 1.5x + 0.5x2 = 10 + -10

Combine like terms: 10 + -10 = 0
-9 + 1.5x + 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'.
-18 + 3x + x2 = 0

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + 3x + 18 + x2 = 0 + 18

Reorder the terms:
-18 + 18 + 3x + x2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + 3x + x2 = 0 + 18
3x + x2 = 0 + 18

Combine like terms: 0 + 18 = 18
3x + x2 = 18

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 18 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 18 + 2.25

Combine like terms: 18 + 2.25 = 20.25
2.25 + 3x + x2 = 20.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 20.25

Calculate the square root of the right side: 4.5

Break this problem into two subproblems by setting 
(x + 1.5) equal to 4.5 and -4.5.

Subproblem 1

x + 1.5 = 4.5 Simplifying x + 1.5 = 4.5 Reorder the terms: 1.5 + x = 4.5 Solving 1.5 + x = 4.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 4.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 4.5 + -1.5 x = 4.5 + -1.5 Combine like terms: 4.5 + -1.5 = 3 x = 3 Simplifying x = 3

Subproblem 2

x + 1.5 = -4.5 Simplifying x + 1.5 = -4.5 Reorder the terms: 1.5 + x = -4.5 Solving 1.5 + x = -4.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -4.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -4.5 + -1.5 x = -4.5 + -1.5 Combine like terms: -4.5 + -1.5 = -6 x = -6 Simplifying x = -6

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3, -6}

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