(a-5)(2a+3)-3(a-4)=50

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Solution for (a-5)(2a+3)-3(a-4)=50 equation:


Simplifying
(a + -5)(2a + 3) + -3(a + -4) = 50

Reorder the terms:
(-5 + a)(2a + 3) + -3(a + -4) = 50

Reorder the terms:
(-5 + a)(3 + 2a) + -3(a + -4) = 50

Multiply (-5 + a) * (3 + 2a)
(-5(3 + 2a) + a(3 + 2a)) + -3(a + -4) = 50
((3 * -5 + 2a * -5) + a(3 + 2a)) + -3(a + -4) = 50
((-15 + -10a) + a(3 + 2a)) + -3(a + -4) = 50
(-15 + -10a + (3 * a + 2a * a)) + -3(a + -4) = 50
(-15 + -10a + (3a + 2a2)) + -3(a + -4) = 50

Combine like terms: -10a + 3a = -7a
(-15 + -7a + 2a2) + -3(a + -4) = 50

Reorder the terms:
-15 + -7a + 2a2 + -3(-4 + a) = 50
-15 + -7a + 2a2 + (-4 * -3 + a * -3) = 50
-15 + -7a + 2a2 + (12 + -3a) = 50

Reorder the terms:
-15 + 12 + -7a + -3a + 2a2 = 50

Combine like terms: -15 + 12 = -3
-3 + -7a + -3a + 2a2 = 50

Combine like terms: -7a + -3a = -10a
-3 + -10a + 2a2 = 50

Solving
-3 + -10a + 2a2 = 50

Solving for variable 'a'.

Reorder the terms:
-3 + -50 + -10a + 2a2 = 50 + -50

Combine like terms: -3 + -50 = -53
-53 + -10a + 2a2 = 50 + -50

Combine like terms: 50 + -50 = 0
-53 + -10a + 2a2 = 0

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

Divide each side by '2'.
-26.5 + -5a + a2 = 0

Move the constant term to the right:

Add '26.5' to each side of the equation.
-26.5 + -5a + 26.5 + a2 = 0 + 26.5

Reorder the terms:
-26.5 + 26.5 + -5a + a2 = 0 + 26.5

Combine like terms: -26.5 + 26.5 = 0.0
0.0 + -5a + a2 = 0 + 26.5
-5a + a2 = 0 + 26.5

Combine like terms: 0 + 26.5 = 26.5
-5a + a2 = 26.5

The a term is -5a.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5a + 6.25 + a2 = 26.5 + 6.25

Reorder the terms:
6.25 + -5a + a2 = 26.5 + 6.25

Combine like terms: 26.5 + 6.25 = 32.75
6.25 + -5a + a2 = 32.75

Factor a perfect square on the left side:
(a + -2.5)(a + -2.5) = 32.75

Calculate the square root of the right side: 5.722761571

Break this problem into two subproblems by setting 
(a + -2.5) equal to 5.722761571 and -5.722761571.

Subproblem 1

a + -2.5 = 5.722761571 Simplifying a + -2.5 = 5.722761571 Reorder the terms: -2.5 + a = 5.722761571 Solving -2.5 + a = 5.722761571 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = 5.722761571 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = 5.722761571 + 2.5 a = 5.722761571 + 2.5 Combine like terms: 5.722761571 + 2.5 = 8.222761571 a = 8.222761571 Simplifying a = 8.222761571

Subproblem 2

a + -2.5 = -5.722761571 Simplifying a + -2.5 = -5.722761571 Reorder the terms: -2.5 + a = -5.722761571 Solving -2.5 + a = -5.722761571 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = -5.722761571 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = -5.722761571 + 2.5 a = -5.722761571 + 2.5 Combine like terms: -5.722761571 + 2.5 = -3.222761571 a = -3.222761571 Simplifying a = -3.222761571

Solution

The solution to the problem is based on the solutions from the subproblems. a = {8.222761571, -3.222761571}

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