(.7a-.21)(.05a+1)=0

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Solution for (.7a-.21)(.05a+1)=0 equation:


Simplifying
(0.7a + -0.21)(0.05a + 1) = 0

Reorder the terms:
(-0.21 + 0.7a)(0.05a + 1) = 0

Reorder the terms:
(-0.21 + 0.7a)(1 + 0.05a) = 0

Multiply (-0.21 + 0.7a) * (1 + 0.05a)
(-0.21(1 + 0.05a) + 0.7a * (1 + 0.05a)) = 0
((1 * -0.21 + 0.05a * -0.21) + 0.7a * (1 + 0.05a)) = 0
((-0.21 + -0.0105a) + 0.7a * (1 + 0.05a)) = 0
(-0.21 + -0.0105a + (1 * 0.7a + 0.05a * 0.7a)) = 0
(-0.21 + -0.0105a + (0.7a + 0.035a2)) = 0

Combine like terms: -0.0105a + 0.7a = 0.6895a
(-0.21 + 0.6895a + 0.035a2) = 0

Solving
-0.21 + 0.6895a + 0.035a2 = 0

Solving for variable 'a'.

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

Divide each side by '0.035'.
-6 + 19.7a + a2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 19.7a + 6 + a2 = 0 + 6

Reorder the terms:
-6 + 6 + 19.7a + a2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 19.7a + a2 = 0 + 6
19.7a + a2 = 0 + 6

Combine like terms: 0 + 6 = 6
19.7a + a2 = 6

The a term is 19.7a.  Take half its coefficient (9.85).
Square it (97.0225) and add it to both sides.

Add '97.0225' to each side of the equation.
19.7a + 97.0225 + a2 = 6 + 97.0225

Reorder the terms:
97.0225 + 19.7a + a2 = 6 + 97.0225

Combine like terms: 6 + 97.0225 = 103.0225
97.0225 + 19.7a + a2 = 103.0225

Factor a perfect square on the left side:
(a + 9.85)(a + 9.85) = 103.0225

Calculate the square root of the right side: 10.15

Break this problem into two subproblems by setting 
(a + 9.85) equal to 10.15 and -10.15.

Subproblem 1

a + 9.85 = 10.15 Simplifying a + 9.85 = 10.15 Reorder the terms: 9.85 + a = 10.15 Solving 9.85 + a = 10.15 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-9.85' to each side of the equation. 9.85 + -9.85 + a = 10.15 + -9.85 Combine like terms: 9.85 + -9.85 = 0.00 0.00 + a = 10.15 + -9.85 a = 10.15 + -9.85 Combine like terms: 10.15 + -9.85 = 0.3 a = 0.3 Simplifying a = 0.3

Subproblem 2

a + 9.85 = -10.15 Simplifying a + 9.85 = -10.15 Reorder the terms: 9.85 + a = -10.15 Solving 9.85 + a = -10.15 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-9.85' to each side of the equation. 9.85 + -9.85 + a = -10.15 + -9.85 Combine like terms: 9.85 + -9.85 = 0.00 0.00 + a = -10.15 + -9.85 a = -10.15 + -9.85 Combine like terms: -10.15 + -9.85 = -20 a = -20 Simplifying a = -20

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.3, -20}

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