2a^2+3a-7=25

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Solution for 2a^2+3a-7=25 equation:


Simplifying
2a2 + 3a + -7 = 25

Reorder the terms:
-7 + 3a + 2a2 = 25

Solving
-7 + 3a + 2a2 = 25

Solving for variable 'a'.

Reorder the terms:
-7 + -25 + 3a + 2a2 = 25 + -25

Combine like terms: -7 + -25 = -32
-32 + 3a + 2a2 = 25 + -25

Combine like terms: 25 + -25 = 0
-32 + 3a + 2a2 = 0

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

Divide each side by '2'.
-16 + 1.5a + a2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + 1.5a + 16 + a2 = 0 + 16

Reorder the terms:
-16 + 16 + 1.5a + a2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + 1.5a + a2 = 0 + 16
1.5a + a2 = 0 + 16

Combine like terms: 0 + 16 = 16
1.5a + a2 = 16

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 = 16 + 0.5625

Reorder the terms:
0.5625 + 1.5a + a2 = 16 + 0.5625

Combine like terms: 16 + 0.5625 = 16.5625
0.5625 + 1.5a + a2 = 16.5625

Factor a perfect square on the left side:
(a + 0.75)(a + 0.75) = 16.5625

Calculate the square root of the right side: 4.069705149

Break this problem into two subproblems by setting 
(a + 0.75) equal to 4.069705149 and -4.069705149.

Subproblem 1

a + 0.75 = 4.069705149 Simplifying a + 0.75 = 4.069705149 Reorder the terms: 0.75 + a = 4.069705149 Solving 0.75 + a = 4.069705149 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 = 4.069705149 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 4.069705149 + -0.75 a = 4.069705149 + -0.75 Combine like terms: 4.069705149 + -0.75 = 3.319705149 a = 3.319705149 Simplifying a = 3.319705149

Subproblem 2

a + 0.75 = -4.069705149 Simplifying a + 0.75 = -4.069705149 Reorder the terms: 0.75 + a = -4.069705149 Solving 0.75 + a = -4.069705149 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 = -4.069705149 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -4.069705149 + -0.75 a = -4.069705149 + -0.75 Combine like terms: -4.069705149 + -0.75 = -4.819705149 a = -4.819705149 Simplifying a = -4.819705149

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.319705149, -4.819705149}

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