a*a+42a-12=0

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Solution for a*a+42a-12=0 equation:


Simplifying
a * a + 42a + -12 = 0

Multiply a * a
a2 + 42a + -12 = 0

Reorder the terms:
-12 + 42a + a2 = 0

Solving
-12 + 42a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 42a + 12 + a2 = 0 + 12

Reorder the terms:
-12 + 12 + 42a + a2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 42a + a2 = 0 + 12
42a + a2 = 0 + 12

Combine like terms: 0 + 12 = 12
42a + a2 = 12

The a term is 42a.  Take half its coefficient (21).
Square it (441) and add it to both sides.

Add '441' to each side of the equation.
42a + 441 + a2 = 12 + 441

Reorder the terms:
441 + 42a + a2 = 12 + 441

Combine like terms: 12 + 441 = 453
441 + 42a + a2 = 453

Factor a perfect square on the left side:
(a + 21)(a + 21) = 453

Calculate the square root of the right side: 21.283796654

Break this problem into two subproblems by setting 
(a + 21) equal to 21.283796654 and -21.283796654.

Subproblem 1

a + 21 = 21.283796654 Simplifying a + 21 = 21.283796654 Reorder the terms: 21 + a = 21.283796654 Solving 21 + a = 21.283796654 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-21' to each side of the equation. 21 + -21 + a = 21.283796654 + -21 Combine like terms: 21 + -21 = 0 0 + a = 21.283796654 + -21 a = 21.283796654 + -21 Combine like terms: 21.283796654 + -21 = 0.283796654 a = 0.283796654 Simplifying a = 0.283796654

Subproblem 2

a + 21 = -21.283796654 Simplifying a + 21 = -21.283796654 Reorder the terms: 21 + a = -21.283796654 Solving 21 + a = -21.283796654 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-21' to each side of the equation. 21 + -21 + a = -21.283796654 + -21 Combine like terms: 21 + -21 = 0 0 + a = -21.283796654 + -21 a = -21.283796654 + -21 Combine like terms: -21.283796654 + -21 = -42.283796654 a = -42.283796654 Simplifying a = -42.283796654

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.283796654, -42.283796654}

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