5a^2-14a-3=10

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Solution for 5a^2-14a-3=10 equation:


Simplifying
5a2 + -14a + -3 = 10

Reorder the terms:
-3 + -14a + 5a2 = 10

Solving
-3 + -14a + 5a2 = 10

Solving for variable 'a'.

Reorder the terms:
-3 + -10 + -14a + 5a2 = 10 + -10

Combine like terms: -3 + -10 = -13
-13 + -14a + 5a2 = 10 + -10

Combine like terms: 10 + -10 = 0
-13 + -14a + 5a2 = 0

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

Divide each side by '5'.
-2.6 + -2.8a + a2 = 0

Move the constant term to the right:

Add '2.6' to each side of the equation.
-2.6 + -2.8a + 2.6 + a2 = 0 + 2.6

Reorder the terms:
-2.6 + 2.6 + -2.8a + a2 = 0 + 2.6

Combine like terms: -2.6 + 2.6 = 0.0
0.0 + -2.8a + a2 = 0 + 2.6
-2.8a + a2 = 0 + 2.6

Combine like terms: 0 + 2.6 = 2.6
-2.8a + a2 = 2.6

The a term is -2.8a.  Take half its coefficient (-1.4).
Square it (1.96) and add it to both sides.

Add '1.96' to each side of the equation.
-2.8a + 1.96 + a2 = 2.6 + 1.96

Reorder the terms:
1.96 + -2.8a + a2 = 2.6 + 1.96

Combine like terms: 2.6 + 1.96 = 4.56
1.96 + -2.8a + a2 = 4.56

Factor a perfect square on the left side:
(a + -1.4)(a + -1.4) = 4.56

Calculate the square root of the right side: 2.13541565

Break this problem into two subproblems by setting 
(a + -1.4) equal to 2.13541565 and -2.13541565.

Subproblem 1

a + -1.4 = 2.13541565 Simplifying a + -1.4 = 2.13541565 Reorder the terms: -1.4 + a = 2.13541565 Solving -1.4 + a = 2.13541565 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.4' to each side of the equation. -1.4 + 1.4 + a = 2.13541565 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + a = 2.13541565 + 1.4 a = 2.13541565 + 1.4 Combine like terms: 2.13541565 + 1.4 = 3.53541565 a = 3.53541565 Simplifying a = 3.53541565

Subproblem 2

a + -1.4 = -2.13541565 Simplifying a + -1.4 = -2.13541565 Reorder the terms: -1.4 + a = -2.13541565 Solving -1.4 + a = -2.13541565 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.4' to each side of the equation. -1.4 + 1.4 + a = -2.13541565 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + a = -2.13541565 + 1.4 a = -2.13541565 + 1.4 Combine like terms: -2.13541565 + 1.4 = -0.73541565 a = -0.73541565 Simplifying a = -0.73541565

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.53541565, -0.73541565}

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