16a^2-24a+1=0

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Solution for 16a^2-24a+1=0 equation:


Simplifying
16a2 + -24a + 1 = 0

Reorder the terms:
1 + -24a + 16a2 = 0

Solving
1 + -24a + 16a2 = 0

Solving for variable 'a'.

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

Divide each side by '16'.
0.0625 + -1.5a + a2 = 0

Move the constant term to the right:

Add '-0.0625' to each side of the equation.
0.0625 + -1.5a + -0.0625 + a2 = 0 + -0.0625

Reorder the terms:
0.0625 + -0.0625 + -1.5a + a2 = 0 + -0.0625

Combine like terms: 0.0625 + -0.0625 = 0.0000
0.0000 + -1.5a + a2 = 0 + -0.0625
-1.5a + a2 = 0 + -0.0625

Combine like terms: 0 + -0.0625 = -0.0625
-1.5a + a2 = -0.0625

The a term is -1.5a.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5a + 0.5625 + a2 = -0.0625 + 0.5625

Reorder the terms:
0.5625 + -1.5a + a2 = -0.0625 + 0.5625

Combine like terms: -0.0625 + 0.5625 = 0.5
0.5625 + -1.5a + a2 = 0.5

Factor a perfect square on the left side:
(a + -0.75)(a + -0.75) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(a + -0.75) equal to 0.707106781 and -0.707106781.

Subproblem 1

a + -0.75 = 0.707106781 Simplifying a + -0.75 = 0.707106781 Reorder the terms: -0.75 + a = 0.707106781 Solving -0.75 + a = 0.707106781 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + a = 0.707106781 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + a = 0.707106781 + 0.75 a = 0.707106781 + 0.75 Combine like terms: 0.707106781 + 0.75 = 1.457106781 a = 1.457106781 Simplifying a = 1.457106781

Subproblem 2

a + -0.75 = -0.707106781 Simplifying a + -0.75 = -0.707106781 Reorder the terms: -0.75 + a = -0.707106781 Solving -0.75 + a = -0.707106781 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + a = -0.707106781 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + a = -0.707106781 + 0.75 a = -0.707106781 + 0.75 Combine like terms: -0.707106781 + 0.75 = 0.042893219 a = 0.042893219 Simplifying a = 0.042893219

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.457106781, 0.042893219}

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