2a^2+3a-7=15

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Solution for 2a^2+3a-7=15 equation:


Simplifying
2a2 + 3a + -7 = 15

Reorder the terms:
-7 + 3a + 2a2 = 15

Solving
-7 + 3a + 2a2 = 15

Solving for variable 'a'.

Reorder the terms:
-7 + -15 + 3a + 2a2 = 15 + -15

Combine like terms: -7 + -15 = -22
-22 + 3a + 2a2 = 15 + -15

Combine like terms: 15 + -15 = 0
-22 + 3a + 2a2 = 0

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

Divide each side by '2'.
-11 + 1.5a + a2 = 0

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + 1.5a + 11 + a2 = 0 + 11

Reorder the terms:
-11 + 11 + 1.5a + a2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + 1.5a + a2 = 0 + 11
1.5a + a2 = 0 + 11

Combine like terms: 0 + 11 = 11
1.5a + a2 = 11

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 = 11 + 0.5625

Reorder the terms:
0.5625 + 1.5a + a2 = 11 + 0.5625

Combine like terms: 11 + 0.5625 = 11.5625
0.5625 + 1.5a + a2 = 11.5625

Factor a perfect square on the left side:
(a + 0.75)(a + 0.75) = 11.5625

Calculate the square root of the right side: 3.400367627

Break this problem into two subproblems by setting 
(a + 0.75) equal to 3.400367627 and -3.400367627.

Subproblem 1

a + 0.75 = 3.400367627 Simplifying a + 0.75 = 3.400367627 Reorder the terms: 0.75 + a = 3.400367627 Solving 0.75 + a = 3.400367627 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 = 3.400367627 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 3.400367627 + -0.75 a = 3.400367627 + -0.75 Combine like terms: 3.400367627 + -0.75 = 2.650367627 a = 2.650367627 Simplifying a = 2.650367627

Subproblem 2

a + 0.75 = -3.400367627 Simplifying a + 0.75 = -3.400367627 Reorder the terms: 0.75 + a = -3.400367627 Solving 0.75 + a = -3.400367627 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 = -3.400367627 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -3.400367627 + -0.75 a = -3.400367627 + -0.75 Combine like terms: -3.400367627 + -0.75 = -4.150367627 a = -4.150367627 Simplifying a = -4.150367627

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.650367627, -4.150367627}

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