84=.5(x)(3+3x)

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Solution for 84=.5(x)(3+3x) equation:


Simplifying
84 = 0.5(x)(3 + 3x)
84 = (3 * 0.5x + 3x * 0.5x)
84 = (1.5x + 1.5x2)

Solving
84 = 1.5x + 1.5x2

Solving for variable 'x'.

Reorder the terms:
84 + -1.5x + -1.5x2 = 1.5x + -1.5x + 1.5x2 + -1.5x2

Combine like terms: 1.5x + -1.5x = 0.0
84 + -1.5x + -1.5x2 = 0.0 + 1.5x2 + -1.5x2
84 + -1.5x + -1.5x2 = 1.5x2 + -1.5x2

Combine like terms: 1.5x2 + -1.5x2 = 0.0
84 + -1.5x + -1.5x2 = 0.0

Begin completing the square.  Divide all terms by
-1.5 the coefficient of the squared term: 

Divide each side by '-1.5'.
-56 + x + x2 = 0

Move the constant term to the right:

Add '56' to each side of the equation.
-56 + x + 56 + x2 = 0 + 56

Reorder the terms:
-56 + 56 + x + x2 = 0 + 56

Combine like terms: -56 + 56 = 0
0 + x + x2 = 0 + 56
x + x2 = 0 + 56

Combine like terms: 0 + 56 = 56
x + x2 = 56

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 56 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 56 + 0.25

Combine like terms: 56 + 0.25 = 56.25
1.25 + x2 = 56.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 56.25

Calculate the square root of the right side: 7.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.5 and -7.5.

Subproblem 1

x + 0.5 = 7.5 Simplifying x + 0.5 = 7.5 Reorder the terms: 0.5 + x = 7.5 Solving 0.5 + x = 7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 7.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.5 + -0.5 x = 7.5 + -0.5 Combine like terms: 7.5 + -0.5 = 7 x = 7 Simplifying x = 7

Subproblem 2

x + 0.5 = -7.5 Simplifying x + 0.5 = -7.5 Reorder the terms: 0.5 + x = -7.5 Solving 0.5 + x = -7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -7.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.5 + -0.5 x = -7.5 + -0.5 Combine like terms: -7.5 + -0.5 = -8 x = -8 Simplifying x = -8

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7, -8}

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