10a^2-17=-9a

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Solution for 10a^2-17=-9a equation:


Simplifying
10a2 + -17 = -9a

Reorder the terms:
-17 + 10a2 = -9a

Solving
-17 + 10a2 = -9a

Solving for variable 'a'.

Reorder the terms:
-17 + 9a + 10a2 = -9a + 9a

Combine like terms: -9a + 9a = 0
-17 + 9a + 10a2 = 0

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

Divide each side by '10'.
-1.7 + 0.9a + a2 = 0

Move the constant term to the right:

Add '1.7' to each side of the equation.
-1.7 + 0.9a + 1.7 + a2 = 0 + 1.7

Reorder the terms:
-1.7 + 1.7 + 0.9a + a2 = 0 + 1.7

Combine like terms: -1.7 + 1.7 = 0.0
0.0 + 0.9a + a2 = 0 + 1.7
0.9a + a2 = 0 + 1.7

Combine like terms: 0 + 1.7 = 1.7
0.9a + a2 = 1.7

The a term is 0.9a.  Take half its coefficient (0.45).
Square it (0.2025) and add it to both sides.

Add '0.2025' to each side of the equation.
0.9a + 0.2025 + a2 = 1.7 + 0.2025

Reorder the terms:
0.2025 + 0.9a + a2 = 1.7 + 0.2025

Combine like terms: 1.7 + 0.2025 = 1.9025
0.2025 + 0.9a + a2 = 1.9025

Factor a perfect square on the left side:
(a + 0.45)(a + 0.45) = 1.9025

Calculate the square root of the right side: 1.379311422

Break this problem into two subproblems by setting 
(a + 0.45) equal to 1.379311422 and -1.379311422.

Subproblem 1

a + 0.45 = 1.379311422 Simplifying a + 0.45 = 1.379311422 Reorder the terms: 0.45 + a = 1.379311422 Solving 0.45 + a = 1.379311422 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + a = 1.379311422 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + a = 1.379311422 + -0.45 a = 1.379311422 + -0.45 Combine like terms: 1.379311422 + -0.45 = 0.929311422 a = 0.929311422 Simplifying a = 0.929311422

Subproblem 2

a + 0.45 = -1.379311422 Simplifying a + 0.45 = -1.379311422 Reorder the terms: 0.45 + a = -1.379311422 Solving 0.45 + a = -1.379311422 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + a = -1.379311422 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + a = -1.379311422 + -0.45 a = -1.379311422 + -0.45 Combine like terms: -1.379311422 + -0.45 = -1.829311422 a = -1.829311422 Simplifying a = -1.829311422

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.929311422, -1.829311422}

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