2a^2+3a-7=40

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Solution for 2a^2+3a-7=40 equation:


Simplifying
2a2 + 3a + -7 = 40

Reorder the terms:
-7 + 3a + 2a2 = 40

Solving
-7 + 3a + 2a2 = 40

Solving for variable 'a'.

Reorder the terms:
-7 + -40 + 3a + 2a2 = 40 + -40

Combine like terms: -7 + -40 = -47
-47 + 3a + 2a2 = 40 + -40

Combine like terms: 40 + -40 = 0
-47 + 3a + 2a2 = 0

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

Divide each side by '2'.
-23.5 + 1.5a + a2 = 0

Move the constant term to the right:

Add '23.5' to each side of the equation.
-23.5 + 1.5a + 23.5 + a2 = 0 + 23.5

Reorder the terms:
-23.5 + 23.5 + 1.5a + a2 = 0 + 23.5

Combine like terms: -23.5 + 23.5 = 0.0
0.0 + 1.5a + a2 = 0 + 23.5
1.5a + a2 = 0 + 23.5

Combine like terms: 0 + 23.5 = 23.5
1.5a + a2 = 23.5

The a term is 1.5a.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5a + 0.5625 + a2 = 23.5 + 0.5625

Reorder the terms:
0.5625 + 1.5a + a2 = 23.5 + 0.5625

Combine like terms: 23.5 + 0.5625 = 24.0625
0.5625 + 1.5a + a2 = 24.0625

Factor a perfect square on the left side:
(a + 0.75)(a + 0.75) = 24.0625

Calculate the square root of the right side: 4.905354218

Break this problem into two subproblems by setting 
(a + 0.75) equal to 4.905354218 and -4.905354218.

Subproblem 1

a + 0.75 = 4.905354218 Simplifying a + 0.75 = 4.905354218 Reorder the terms: 0.75 + a = 4.905354218 Solving 0.75 + a = 4.905354218 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 4.905354218 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 4.905354218 + -0.75 a = 4.905354218 + -0.75 Combine like terms: 4.905354218 + -0.75 = 4.155354218 a = 4.155354218 Simplifying a = 4.155354218

Subproblem 2

a + 0.75 = -4.905354218 Simplifying a + 0.75 = -4.905354218 Reorder the terms: 0.75 + a = -4.905354218 Solving 0.75 + a = -4.905354218 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -4.905354218 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -4.905354218 + -0.75 a = -4.905354218 + -0.75 Combine like terms: -4.905354218 + -0.75 = -5.655354218 a = -5.655354218 Simplifying a = -5.655354218

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.155354218, -5.655354218}

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