5a^2-19a=15

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Solution for 5a^2-19a=15 equation:


Simplifying
5a2 + -19a = 15

Reorder the terms:
-19a + 5a2 = 15

Solving
-19a + 5a2 = 15

Solving for variable 'a'.

Reorder the terms:
-15 + -19a + 5a2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + -19a + 5a2 = 0

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

Divide each side by '5'.
-3 + -3.8a + a2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -3.8a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + -3.8a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -3.8a + a2 = 0 + 3
-3.8a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
-3.8a + a2 = 3

The a term is -3.8a.  Take half its coefficient (-1.9).
Square it (3.61) and add it to both sides.

Add '3.61' to each side of the equation.
-3.8a + 3.61 + a2 = 3 + 3.61

Reorder the terms:
3.61 + -3.8a + a2 = 3 + 3.61

Combine like terms: 3 + 3.61 = 6.61
3.61 + -3.8a + a2 = 6.61

Factor a perfect square on the left side:
(a + -1.9)(a + -1.9) = 6.61

Calculate the square root of the right side: 2.570992026

Break this problem into two subproblems by setting 
(a + -1.9) equal to 2.570992026 and -2.570992026.

Subproblem 1

a + -1.9 = 2.570992026 Simplifying a + -1.9 = 2.570992026 Reorder the terms: -1.9 + a = 2.570992026 Solving -1.9 + a = 2.570992026 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.9' to each side of the equation. -1.9 + 1.9 + a = 2.570992026 + 1.9 Combine like terms: -1.9 + 1.9 = 0.0 0.0 + a = 2.570992026 + 1.9 a = 2.570992026 + 1.9 Combine like terms: 2.570992026 + 1.9 = 4.470992026 a = 4.470992026 Simplifying a = 4.470992026

Subproblem 2

a + -1.9 = -2.570992026 Simplifying a + -1.9 = -2.570992026 Reorder the terms: -1.9 + a = -2.570992026 Solving -1.9 + a = -2.570992026 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.9' to each side of the equation. -1.9 + 1.9 + a = -2.570992026 + 1.9 Combine like terms: -1.9 + 1.9 = 0.0 0.0 + a = -2.570992026 + 1.9 a = -2.570992026 + 1.9 Combine like terms: -2.570992026 + 1.9 = -0.670992026 a = -0.670992026 Simplifying a = -0.670992026

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.470992026, -0.670992026}

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