7a(3a-4)=12

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Solution for 7a(3a-4)=12 equation:


Simplifying
7a(3a + -4) = 12

Reorder the terms:
7a(-4 + 3a) = 12
(-4 * 7a + 3a * 7a) = 12
(-28a + 21a2) = 12

Solving
-28a + 21a2 = 12

Solving for variable 'a'.

Reorder the terms:
-12 + -28a + 21a2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + -28a + 21a2 = 0

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

Divide each side by '21'.
-0.5714285714 + -1.333333333a + a2 = 0

Move the constant term to the right:

Add '0.5714285714' to each side of the equation.
-0.5714285714 + -1.333333333a + 0.5714285714 + a2 = 0 + 0.5714285714

Reorder the terms:
-0.5714285714 + 0.5714285714 + -1.333333333a + a2 = 0 + 0.5714285714

Combine like terms: -0.5714285714 + 0.5714285714 = 0.0000000000
0.0000000000 + -1.333333333a + a2 = 0 + 0.5714285714
-1.333333333a + a2 = 0 + 0.5714285714

Combine like terms: 0 + 0.5714285714 = 0.5714285714
-1.333333333a + a2 = 0.5714285714

The a term is -1.333333333a.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333a + 0.4444444442 + a2 = 0.5714285714 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333a + a2 = 0.5714285714 + 0.4444444442

Combine like terms: 0.5714285714 + 0.4444444442 = 1.0158730156
0.4444444442 + -1.333333333a + a2 = 1.0158730156

Factor a perfect square on the left side:
(a + -0.6666666665)(a + -0.6666666665) = 1.0158730156

Calculate the square root of the right side: 1.007905261

Break this problem into two subproblems by setting 
(a + -0.6666666665) equal to 1.007905261 and -1.007905261.

Subproblem 1

a + -0.6666666665 = 1.007905261 Simplifying a + -0.6666666665 = 1.007905261 Reorder the terms: -0.6666666665 + a = 1.007905261 Solving -0.6666666665 + a = 1.007905261 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + a = 1.007905261 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + a = 1.007905261 + 0.6666666665 a = 1.007905261 + 0.6666666665 Combine like terms: 1.007905261 + 0.6666666665 = 1.6745719275 a = 1.6745719275 Simplifying a = 1.6745719275

Subproblem 2

a + -0.6666666665 = -1.007905261 Simplifying a + -0.6666666665 = -1.007905261 Reorder the terms: -0.6666666665 + a = -1.007905261 Solving -0.6666666665 + a = -1.007905261 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + a = -1.007905261 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + a = -1.007905261 + 0.6666666665 a = -1.007905261 + 0.6666666665 Combine like terms: -1.007905261 + 0.6666666665 = -0.3412385945 a = -0.3412385945 Simplifying a = -0.3412385945

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.6745719275, -0.3412385945}

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