(2x+5)(x-2)=78

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Solution for (2x+5)(x-2)=78 equation:


Simplifying
(2x + 5)(x + -2) = 78

Reorder the terms:
(5 + 2x)(x + -2) = 78

Reorder the terms:
(5 + 2x)(-2 + x) = 78

Multiply (5 + 2x) * (-2 + x)
(5(-2 + x) + 2x * (-2 + x)) = 78
((-2 * 5 + x * 5) + 2x * (-2 + x)) = 78
((-10 + 5x) + 2x * (-2 + x)) = 78
(-10 + 5x + (-2 * 2x + x * 2x)) = 78
(-10 + 5x + (-4x + 2x2)) = 78

Combine like terms: 5x + -4x = 1x
(-10 + 1x + 2x2) = 78

Solving
-10 + 1x + 2x2 = 78

Solving for variable 'x'.

Reorder the terms:
-10 + -78 + 1x + 2x2 = 78 + -78

Combine like terms: -10 + -78 = -88
-88 + 1x + 2x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-88 + 1x + 2x2 = 0

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

Divide each side by '2'.
-44 + 0.5x + x2 = 0

Move the constant term to the right:

Add '44' to each side of the equation.
-44 + 0.5x + 44 + x2 = 0 + 44

Reorder the terms:
-44 + 44 + 0.5x + x2 = 0 + 44

Combine like terms: -44 + 44 = 0
0 + 0.5x + x2 = 0 + 44
0.5x + x2 = 0 + 44

Combine like terms: 0 + 44 = 44
0.5x + x2 = 44

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 44 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 44 + 0.0625

Combine like terms: 44 + 0.0625 = 44.0625
0.0625 + 0.5x + x2 = 44.0625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 44.0625

Calculate the square root of the right side: 6.637959024

Break this problem into two subproblems by setting 
(x + 0.25) equal to 6.637959024 and -6.637959024.

Subproblem 1

x + 0.25 = 6.637959024 Simplifying x + 0.25 = 6.637959024 Reorder the terms: 0.25 + x = 6.637959024 Solving 0.25 + x = 6.637959024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 6.637959024 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 6.637959024 + -0.25 x = 6.637959024 + -0.25 Combine like terms: 6.637959024 + -0.25 = 6.387959024 x = 6.387959024 Simplifying x = 6.387959024

Subproblem 2

x + 0.25 = -6.637959024 Simplifying x + 0.25 = -6.637959024 Reorder the terms: 0.25 + x = -6.637959024 Solving 0.25 + x = -6.637959024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -6.637959024 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -6.637959024 + -0.25 x = -6.637959024 + -0.25 Combine like terms: -6.637959024 + -0.25 = -6.887959024 x = -6.887959024 Simplifying x = -6.887959024

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.387959024, -6.887959024}

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