.5x(x+3)=10

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Solution for .5x(x+3)=10 equation:


Simplifying
0.5x(x + 3) = 10

Reorder the terms:
0.5x(3 + x) = 10
(3 * 0.5x + x * 0.5x) = 10
(1.5x + 0.5x2) = 10

Solving
1.5x + 0.5x2 = 10

Solving for variable 'x'.

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

Combine like terms: 10 + -10 = 0
-10 + 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'.
-20 + 3x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 3x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + 3x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 3x + x2 = 0 + 20
3x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
3x + x2 = 20

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 = 20 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 20 + 2.25

Combine like terms: 20 + 2.25 = 22.25
2.25 + 3x + x2 = 22.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(x + 1.5) equal to 4.716990566 and -4.716990566.

Subproblem 1

x + 1.5 = 4.716990566 Simplifying x + 1.5 = 4.716990566 Reorder the terms: 1.5 + x = 4.716990566 Solving 1.5 + x = 4.716990566 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 = 4.716990566 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 4.716990566 + -1.5 x = 4.716990566 + -1.5 Combine like terms: 4.716990566 + -1.5 = 3.216990566 x = 3.216990566 Simplifying x = 3.216990566

Subproblem 2

x + 1.5 = -4.716990566 Simplifying x + 1.5 = -4.716990566 Reorder the terms: 1.5 + x = -4.716990566 Solving 1.5 + x = -4.716990566 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 = -4.716990566 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -4.716990566 + -1.5 x = -4.716990566 + -1.5 Combine like terms: -4.716990566 + -1.5 = -6.216990566 x = -6.216990566 Simplifying x = -6.216990566

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.216990566, -6.216990566}

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