2a^2+3a-7=100

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Solution for 2a^2+3a-7=100 equation:


Simplifying
2a2 + 3a + -7 = 100

Reorder the terms:
-7 + 3a + 2a2 = 100

Solving
-7 + 3a + 2a2 = 100

Solving for variable 'a'.

Reorder the terms:
-7 + -100 + 3a + 2a2 = 100 + -100

Combine like terms: -7 + -100 = -107
-107 + 3a + 2a2 = 100 + -100

Combine like terms: 100 + -100 = 0
-107 + 3a + 2a2 = 0

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

Divide each side by '2'.
-53.5 + 1.5a + a2 = 0

Move the constant term to the right:

Add '53.5' to each side of the equation.
-53.5 + 1.5a + 53.5 + a2 = 0 + 53.5

Reorder the terms:
-53.5 + 53.5 + 1.5a + a2 = 0 + 53.5

Combine like terms: -53.5 + 53.5 = 0.0
0.0 + 1.5a + a2 = 0 + 53.5
1.5a + a2 = 0 + 53.5

Combine like terms: 0 + 53.5 = 53.5
1.5a + a2 = 53.5

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 = 53.5 + 0.5625

Reorder the terms:
0.5625 + 1.5a + a2 = 53.5 + 0.5625

Combine like terms: 53.5 + 0.5625 = 54.0625
0.5625 + 1.5a + a2 = 54.0625

Factor a perfect square on the left side:
(a + 0.75)(a + 0.75) = 54.0625

Calculate the square root of the right side: 7.352720585

Break this problem into two subproblems by setting 
(a + 0.75) equal to 7.352720585 and -7.352720585.

Subproblem 1

a + 0.75 = 7.352720585 Simplifying a + 0.75 = 7.352720585 Reorder the terms: 0.75 + a = 7.352720585 Solving 0.75 + a = 7.352720585 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 = 7.352720585 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 7.352720585 + -0.75 a = 7.352720585 + -0.75 Combine like terms: 7.352720585 + -0.75 = 6.602720585 a = 6.602720585 Simplifying a = 6.602720585

Subproblem 2

a + 0.75 = -7.352720585 Simplifying a + 0.75 = -7.352720585 Reorder the terms: 0.75 + a = -7.352720585 Solving 0.75 + a = -7.352720585 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 = -7.352720585 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -7.352720585 + -0.75 a = -7.352720585 + -0.75 Combine like terms: -7.352720585 + -0.75 = -8.102720585 a = -8.102720585 Simplifying a = -8.102720585

Solution

The solution to the problem is based on the solutions from the subproblems. a = {6.602720585, -8.102720585}

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