4a^2+5a-14=0

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Solution for 4a^2+5a-14=0 equation:


Simplifying
4a2 + 5a + -14 = 0

Reorder the terms:
-14 + 5a + 4a2 = 0

Solving
-14 + 5a + 4a2 = 0

Solving for variable 'a'.

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

Divide each side by '4'.
-3.5 + 1.25a + a2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 1.25a + 3.5 + a2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 1.25a + a2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 1.25a + a2 = 0 + 3.5
1.25a + a2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
1.25a + a2 = 3.5

The a term is 1.25a.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25a + 0.390625 + a2 = 3.5 + 0.390625

Reorder the terms:
0.390625 + 1.25a + a2 = 3.5 + 0.390625

Combine like terms: 3.5 + 0.390625 = 3.890625
0.390625 + 1.25a + a2 = 3.890625

Factor a perfect square on the left side:
(a + 0.625)(a + 0.625) = 3.890625

Calculate the square root of the right side: 1.97246673

Break this problem into two subproblems by setting 
(a + 0.625) equal to 1.97246673 and -1.97246673.

Subproblem 1

a + 0.625 = 1.97246673 Simplifying a + 0.625 = 1.97246673 Reorder the terms: 0.625 + a = 1.97246673 Solving 0.625 + a = 1.97246673 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + a = 1.97246673 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + a = 1.97246673 + -0.625 a = 1.97246673 + -0.625 Combine like terms: 1.97246673 + -0.625 = 1.34746673 a = 1.34746673 Simplifying a = 1.34746673

Subproblem 2

a + 0.625 = -1.97246673 Simplifying a + 0.625 = -1.97246673 Reorder the terms: 0.625 + a = -1.97246673 Solving 0.625 + a = -1.97246673 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + a = -1.97246673 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + a = -1.97246673 + -0.625 a = -1.97246673 + -0.625 Combine like terms: -1.97246673 + -0.625 = -2.59746673 a = -2.59746673 Simplifying a = -2.59746673

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.34746673, -2.59746673}

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