(0.5)(x^2+7x)=49

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Solution for (0.5)(x^2+7x)=49 equation:


Simplifying
(0.5)(x2 + 7x) = 49

Reorder the terms:
0.5(7x + x2) = 49
(7x * 0.5 + x2 * 0.5) = 49
(3.5x + 0.5x2) = 49

Solving
3.5x + 0.5x2 = 49

Solving for variable 'x'.

Reorder the terms:
-49 + 3.5x + 0.5x2 = 49 + -49

Combine like terms: 49 + -49 = 0
-49 + 3.5x + 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'.
-98 + 7x + x2 = 0

Move the constant term to the right:

Add '98' to each side of the equation.
-98 + 7x + 98 + x2 = 0 + 98

Reorder the terms:
-98 + 98 + 7x + x2 = 0 + 98

Combine like terms: -98 + 98 = 0
0 + 7x + x2 = 0 + 98
7x + x2 = 0 + 98

Combine like terms: 0 + 98 = 98
7x + x2 = 98

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = 98 + 12.25

Reorder the terms:
12.25 + 7x + x2 = 98 + 12.25

Combine like terms: 98 + 12.25 = 110.25
12.25 + 7x + x2 = 110.25

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 110.25

Calculate the square root of the right side: 10.5

Break this problem into two subproblems by setting 
(x + 3.5) equal to 10.5 and -10.5.

Subproblem 1

x + 3.5 = 10.5 Simplifying x + 3.5 = 10.5 Reorder the terms: 3.5 + x = 10.5 Solving 3.5 + x = 10.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 10.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 10.5 + -3.5 x = 10.5 + -3.5 Combine like terms: 10.5 + -3.5 = 7 x = 7 Simplifying x = 7

Subproblem 2

x + 3.5 = -10.5 Simplifying x + 3.5 = -10.5 Reorder the terms: 3.5 + x = -10.5 Solving 3.5 + x = -10.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -10.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -10.5 + -3.5 x = -10.5 + -3.5 Combine like terms: -10.5 + -3.5 = -14 x = -14 Simplifying x = -14

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7, -14}

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