4a^2-1a+5=15

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Solution for 4a^2-1a+5=15 equation:


Simplifying
4a2 + -1a + 5 = 15

Reorder the terms:
5 + -1a + 4a2 = 15

Solving
5 + -1a + 4a2 = 15

Solving for variable 'a'.

Reorder the terms:
5 + -15 + -1a + 4a2 = 15 + -15

Combine like terms: 5 + -15 = -10
-10 + -1a + 4a2 = 15 + -15

Combine like terms: 15 + -15 = 0
-10 + -1a + 4a2 = 0

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

Divide each side by '4'.
-2.5 + -0.25a + a2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -0.25a + 2.5 + a2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -0.25a + a2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -0.25a + a2 = 0 + 2.5
-0.25a + a2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-0.25a + a2 = 2.5

The a term is -0.25a.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25a + 0.015625 + a2 = 2.5 + 0.015625

Reorder the terms:
0.015625 + -0.25a + a2 = 2.5 + 0.015625

Combine like terms: 2.5 + 0.015625 = 2.515625
0.015625 + -0.25a + a2 = 2.515625

Factor a perfect square on the left side:
(a + -0.125)(a + -0.125) = 2.515625

Calculate the square root of the right side: 1.586072193

Break this problem into two subproblems by setting 
(a + -0.125) equal to 1.586072193 and -1.586072193.

Subproblem 1

a + -0.125 = 1.586072193 Simplifying a + -0.125 = 1.586072193 Reorder the terms: -0.125 + a = 1.586072193 Solving -0.125 + a = 1.586072193 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = 1.586072193 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = 1.586072193 + 0.125 a = 1.586072193 + 0.125 Combine like terms: 1.586072193 + 0.125 = 1.711072193 a = 1.711072193 Simplifying a = 1.711072193

Subproblem 2

a + -0.125 = -1.586072193 Simplifying a + -0.125 = -1.586072193 Reorder the terms: -0.125 + a = -1.586072193 Solving -0.125 + a = -1.586072193 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = -1.586072193 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = -1.586072193 + 0.125 a = -1.586072193 + 0.125 Combine like terms: -1.586072193 + 0.125 = -1.461072193 a = -1.461072193 Simplifying a = -1.461072193

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.711072193, -1.461072193}

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