9a^2=30a+25

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Solution for 9a^2=30a+25 equation:


Simplifying
9a2 = 30a + 25

Reorder the terms:
9a2 = 25 + 30a

Solving
9a2 = 25 + 30a

Solving for variable 'a'.

Reorder the terms:
-25 + -30a + 9a2 = 25 + 30a + -25 + -30a

Reorder the terms:
-25 + -30a + 9a2 = 25 + -25 + 30a + -30a

Combine like terms: 25 + -25 = 0
-25 + -30a + 9a2 = 0 + 30a + -30a
-25 + -30a + 9a2 = 30a + -30a

Combine like terms: 30a + -30a = 0
-25 + -30a + 9a2 = 0

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

Divide each side by '9'.
-2.777777778 + -3.333333333a + a2 = 0

Move the constant term to the right:

Add '2.777777778' to each side of the equation.
-2.777777778 + -3.333333333a + 2.777777778 + a2 = 0 + 2.777777778

Reorder the terms:
-2.777777778 + 2.777777778 + -3.333333333a + a2 = 0 + 2.777777778

Combine like terms: -2.777777778 + 2.777777778 = 0.000000000
0.000000000 + -3.333333333a + a2 = 0 + 2.777777778
-3.333333333a + a2 = 0 + 2.777777778

Combine like terms: 0 + 2.777777778 = 2.777777778
-3.333333333a + a2 = 2.777777778

The a term is -3.333333333a.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333a + 2.777777779 + a2 = 2.777777778 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333a + a2 = 2.777777778 + 2.777777779

Combine like terms: 2.777777778 + 2.777777779 = 5.555555557
2.777777779 + -3.333333333a + a2 = 5.555555557

Factor a perfect square on the left side:
(a + -1.666666667)(a + -1.666666667) = 5.555555557

Calculate the square root of the right side: 2.357022604

Break this problem into two subproblems by setting 
(a + -1.666666667) equal to 2.357022604 and -2.357022604.

Subproblem 1

a + -1.666666667 = 2.357022604 Simplifying a + -1.666666667 = 2.357022604 Reorder the terms: -1.666666667 + a = 2.357022604 Solving -1.666666667 + a = 2.357022604 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + a = 2.357022604 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = 2.357022604 + 1.666666667 a = 2.357022604 + 1.666666667 Combine like terms: 2.357022604 + 1.666666667 = 4.023689271 a = 4.023689271 Simplifying a = 4.023689271

Subproblem 2

a + -1.666666667 = -2.357022604 Simplifying a + -1.666666667 = -2.357022604 Reorder the terms: -1.666666667 + a = -2.357022604 Solving -1.666666667 + a = -2.357022604 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + a = -2.357022604 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = -2.357022604 + 1.666666667 a = -2.357022604 + 1.666666667 Combine like terms: -2.357022604 + 1.666666667 = -0.690355937 a = -0.690355937 Simplifying a = -0.690355937

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.023689271, -0.690355937}

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