4a^2-9a=25

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Solution for 4a^2-9a=25 equation:


Simplifying
4a2 + -9a = 25

Reorder the terms:
-9a + 4a2 = 25

Solving
-9a + 4a2 = 25

Solving for variable 'a'.

Reorder the terms:
-25 + -9a + 4a2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + -9a + 4a2 = 0

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

Divide each side by '4'.
-6.25 + -2.25a + a2 = 0

Move the constant term to the right:

Add '6.25' to each side of the equation.
-6.25 + -2.25a + 6.25 + a2 = 0 + 6.25

Reorder the terms:
-6.25 + 6.25 + -2.25a + a2 = 0 + 6.25

Combine like terms: -6.25 + 6.25 = 0.00
0.00 + -2.25a + a2 = 0 + 6.25
-2.25a + a2 = 0 + 6.25

Combine like terms: 0 + 6.25 = 6.25
-2.25a + a2 = 6.25

The a term is -2.25a.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25a + 1.265625 + a2 = 6.25 + 1.265625

Reorder the terms:
1.265625 + -2.25a + a2 = 6.25 + 1.265625

Combine like terms: 6.25 + 1.265625 = 7.515625
1.265625 + -2.25a + a2 = 7.515625

Factor a perfect square on the left side:
(a + -1.125)(a + -1.125) = 7.515625

Calculate the square root of the right side: 2.741464025

Break this problem into two subproblems by setting 
(a + -1.125) equal to 2.741464025 and -2.741464025.

Subproblem 1

a + -1.125 = 2.741464025 Simplifying a + -1.125 = 2.741464025 Reorder the terms: -1.125 + a = 2.741464025 Solving -1.125 + a = 2.741464025 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + a = 2.741464025 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + a = 2.741464025 + 1.125 a = 2.741464025 + 1.125 Combine like terms: 2.741464025 + 1.125 = 3.866464025 a = 3.866464025 Simplifying a = 3.866464025

Subproblem 2

a + -1.125 = -2.741464025 Simplifying a + -1.125 = -2.741464025 Reorder the terms: -1.125 + a = -2.741464025 Solving -1.125 + a = -2.741464025 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + a = -2.741464025 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + a = -2.741464025 + 1.125 a = -2.741464025 + 1.125 Combine like terms: -2.741464025 + 1.125 = -1.616464025 a = -1.616464025 Simplifying a = -1.616464025

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.866464025, -1.616464025}

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