7-a^2=3a

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


Simplifying
7 + -1a2 = 3a

Solving
7 + -1a2 = 3a

Solving for variable 'a'.

Reorder the terms:
7 + -3a + -1a2 = 3a + -3a

Combine like terms: 3a + -3a = 0
7 + -3a + -1a2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-7 + 3a + a2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 3a + 7 + a2 = 0 + 7

Reorder the terms:
-7 + 7 + 3a + a2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 3a + a2 = 0 + 7
3a + a2 = 0 + 7

Combine like terms: 0 + 7 = 7
3a + a2 = 7

The a term is 3a.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3a + 2.25 + a2 = 7 + 2.25

Reorder the terms:
2.25 + 3a + a2 = 7 + 2.25

Combine like terms: 7 + 2.25 = 9.25
2.25 + 3a + a2 = 9.25

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(a + 1.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

a + 1.5 = 3.041381265 Simplifying a + 1.5 = 3.041381265 Reorder the terms: 1.5 + a = 3.041381265 Solving 1.5 + a = 3.041381265 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 3.041381265 + -1.5 a = 3.041381265 + -1.5 Combine like terms: 3.041381265 + -1.5 = 1.541381265 a = 1.541381265 Simplifying a = 1.541381265

Subproblem 2

a + 1.5 = -3.041381265 Simplifying a + 1.5 = -3.041381265 Reorder the terms: 1.5 + a = -3.041381265 Solving 1.5 + a = -3.041381265 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -3.041381265 + -1.5 a = -3.041381265 + -1.5 Combine like terms: -3.041381265 + -1.5 = -4.541381265 a = -4.541381265 Simplifying a = -4.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.541381265, -4.541381265}

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