x(2x+7)=75

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Solution for x(2x+7)=75 equation:


Simplifying
x(2x + 7) = 75

Reorder the terms:
x(7 + 2x) = 75
(7 * x + 2x * x) = 75
(7x + 2x2) = 75

Solving
7x + 2x2 = 75

Solving for variable 'x'.

Reorder the terms:
-75 + 7x + 2x2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + 7x + 2x2 = 0

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

Divide each side by '2'.
-37.5 + 3.5x + x2 = 0

Move the constant term to the right:

Add '37.5' to each side of the equation.
-37.5 + 3.5x + 37.5 + x2 = 0 + 37.5

Reorder the terms:
-37.5 + 37.5 + 3.5x + x2 = 0 + 37.5

Combine like terms: -37.5 + 37.5 = 0.0
0.0 + 3.5x + x2 = 0 + 37.5
3.5x + x2 = 0 + 37.5

Combine like terms: 0 + 37.5 = 37.5
3.5x + x2 = 37.5

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = 37.5 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = 37.5 + 3.0625

Combine like terms: 37.5 + 3.0625 = 40.5625
3.0625 + 3.5x + x2 = 40.5625

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 40.5625

Calculate the square root of the right side: 6.368869601

Break this problem into two subproblems by setting 
(x + 1.75) equal to 6.368869601 and -6.368869601.

Subproblem 1

x + 1.75 = 6.368869601 Simplifying x + 1.75 = 6.368869601 Reorder the terms: 1.75 + x = 6.368869601 Solving 1.75 + x = 6.368869601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 6.368869601 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 6.368869601 + -1.75 x = 6.368869601 + -1.75 Combine like terms: 6.368869601 + -1.75 = 4.618869601 x = 4.618869601 Simplifying x = 4.618869601

Subproblem 2

x + 1.75 = -6.368869601 Simplifying x + 1.75 = -6.368869601 Reorder the terms: 1.75 + x = -6.368869601 Solving 1.75 + x = -6.368869601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -6.368869601 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -6.368869601 + -1.75 x = -6.368869601 + -1.75 Combine like terms: -6.368869601 + -1.75 = -8.118869601 x = -8.118869601 Simplifying x = -8.118869601

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.618869601, -8.118869601}

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