4a^2+3a=2

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Solution for 4a^2+3a=2 equation:


Simplifying
4a2 + 3a = 2

Reorder the terms:
3a + 4a2 = 2

Solving
3a + 4a2 = 2

Solving for variable 'a'.

Reorder the terms:
-2 + 3a + 4a2 = 2 + -2

Combine like terms: 2 + -2 = 0
-2 + 3a + 4a2 = 0

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

Divide each side by '4'.
-0.5 + 0.75a + a2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 0.75a + 0.5 + a2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 0.75a + a2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 0.75a + a2 = 0 + 0.5
0.75a + a2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
0.75a + a2 = 0.5

The a term is 0.75a.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75a + 0.140625 + a2 = 0.5 + 0.140625

Reorder the terms:
0.140625 + 0.75a + a2 = 0.5 + 0.140625

Combine like terms: 0.5 + 0.140625 = 0.640625
0.140625 + 0.75a + a2 = 0.640625

Factor a perfect square on the left side:
(a + 0.375)(a + 0.375) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(a + 0.375) equal to 0.80039053 and -0.80039053.

Subproblem 1

a + 0.375 = 0.80039053 Simplifying a + 0.375 = 0.80039053 Reorder the terms: 0.375 + a = 0.80039053 Solving 0.375 + a = 0.80039053 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + a = 0.80039053 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + a = 0.80039053 + -0.375 a = 0.80039053 + -0.375 Combine like terms: 0.80039053 + -0.375 = 0.42539053 a = 0.42539053 Simplifying a = 0.42539053

Subproblem 2

a + 0.375 = -0.80039053 Simplifying a + 0.375 = -0.80039053 Reorder the terms: 0.375 + a = -0.80039053 Solving 0.375 + a = -0.80039053 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + a = -0.80039053 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + a = -0.80039053 + -0.375 a = -0.80039053 + -0.375 Combine like terms: -0.80039053 + -0.375 = -1.17539053 a = -1.17539053 Simplifying a = -1.17539053

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.42539053, -1.17539053}

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