a^2+3a-21=0

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Solution for a^2+3a-21=0 equation:


Simplifying
a2 + 3a + -21 = 0

Reorder the terms:
-21 + 3a + a2 = 0

Solving
-21 + 3a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 3a + 21 + a2 = 0 + 21

Reorder the terms:
-21 + 21 + 3a + a2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 3a + a2 = 0 + 21
3a + a2 = 0 + 21

Combine like terms: 0 + 21 = 21
3a + a2 = 21

The a term is 3a.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3a + 2.25 + a2 = 21 + 2.25

Reorder the terms:
2.25 + 3a + a2 = 21 + 2.25

Combine like terms: 21 + 2.25 = 23.25
2.25 + 3a + a2 = 23.25

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 23.25

Calculate the square root of the right side: 4.82182538

Break this problem into two subproblems by setting 
(a + 1.5) equal to 4.82182538 and -4.82182538.

Subproblem 1

a + 1.5 = 4.82182538 Simplifying a + 1.5 = 4.82182538 Reorder the terms: 1.5 + a = 4.82182538 Solving 1.5 + a = 4.82182538 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 4.82182538 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 4.82182538 + -1.5 a = 4.82182538 + -1.5 Combine like terms: 4.82182538 + -1.5 = 3.32182538 a = 3.32182538 Simplifying a = 3.32182538

Subproblem 2

a + 1.5 = -4.82182538 Simplifying a + 1.5 = -4.82182538 Reorder the terms: 1.5 + a = -4.82182538 Solving 1.5 + a = -4.82182538 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -4.82182538 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -4.82182538 + -1.5 a = -4.82182538 + -1.5 Combine like terms: -4.82182538 + -1.5 = -6.32182538 a = -6.32182538 Simplifying a = -6.32182538

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.32182538, -6.32182538}

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