4a^2-14a-21=0

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Solution for 4a^2-14a-21=0 equation:


Simplifying
4a2 + -14a + -21 = 0

Reorder the terms:
-21 + -14a + 4a2 = 0

Solving
-21 + -14a + 4a2 = 0

Solving for variable 'a'.

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

Divide each side by '4'.
-5.25 + -3.5a + a2 = 0

Move the constant term to the right:

Add '5.25' to each side of the equation.
-5.25 + -3.5a + 5.25 + a2 = 0 + 5.25

Reorder the terms:
-5.25 + 5.25 + -3.5a + a2 = 0 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + -3.5a + a2 = 0 + 5.25
-3.5a + a2 = 0 + 5.25

Combine like terms: 0 + 5.25 = 5.25
-3.5a + a2 = 5.25

The a term is -3.5a.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5a + 3.0625 + a2 = 5.25 + 3.0625

Reorder the terms:
3.0625 + -3.5a + a2 = 5.25 + 3.0625

Combine like terms: 5.25 + 3.0625 = 8.3125
3.0625 + -3.5a + a2 = 8.3125

Factor a perfect square on the left side:
(a + -1.75)(a + -1.75) = 8.3125

Calculate the square root of the right side: 2.883140649

Break this problem into two subproblems by setting 
(a + -1.75) equal to 2.883140649 and -2.883140649.

Subproblem 1

a + -1.75 = 2.883140649 Simplifying a + -1.75 = 2.883140649 Reorder the terms: -1.75 + a = 2.883140649 Solving -1.75 + a = 2.883140649 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = 2.883140649 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = 2.883140649 + 1.75 a = 2.883140649 + 1.75 Combine like terms: 2.883140649 + 1.75 = 4.633140649 a = 4.633140649 Simplifying a = 4.633140649

Subproblem 2

a + -1.75 = -2.883140649 Simplifying a + -1.75 = -2.883140649 Reorder the terms: -1.75 + a = -2.883140649 Solving -1.75 + a = -2.883140649 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = -2.883140649 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = -2.883140649 + 1.75 a = -2.883140649 + 1.75 Combine like terms: -2.883140649 + 1.75 = -1.133140649 a = -1.133140649 Simplifying a = -1.133140649

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.633140649, -1.133140649}

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