q^2+5q-102=0

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Solution for q^2+5q-102=0 equation:


Simplifying
q2 + 5q + -102 = 0

Reorder the terms:
-102 + 5q + q2 = 0

Solving
-102 + 5q + q2 = 0

Solving for variable 'q'.

Begin completing the square.

Move the constant term to the right:

Add '102' to each side of the equation.
-102 + 5q + 102 + q2 = 0 + 102

Reorder the terms:
-102 + 102 + 5q + q2 = 0 + 102

Combine like terms: -102 + 102 = 0
0 + 5q + q2 = 0 + 102
5q + q2 = 0 + 102

Combine like terms: 0 + 102 = 102
5q + q2 = 102

The q term is 5q.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5q + 6.25 + q2 = 102 + 6.25

Reorder the terms:
6.25 + 5q + q2 = 102 + 6.25

Combine like terms: 102 + 6.25 = 108.25
6.25 + 5q + q2 = 108.25

Factor a perfect square on the left side:
(q + 2.5)(q + 2.5) = 108.25

Calculate the square root of the right side: 10.404326023

Break this problem into two subproblems by setting 
(q + 2.5) equal to 10.404326023 and -10.404326023.

Subproblem 1

q + 2.5 = 10.404326023 Simplifying q + 2.5 = 10.404326023 Reorder the terms: 2.5 + q = 10.404326023 Solving 2.5 + q = 10.404326023 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + q = 10.404326023 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + q = 10.404326023 + -2.5 q = 10.404326023 + -2.5 Combine like terms: 10.404326023 + -2.5 = 7.904326023 q = 7.904326023 Simplifying q = 7.904326023

Subproblem 2

q + 2.5 = -10.404326023 Simplifying q + 2.5 = -10.404326023 Reorder the terms: 2.5 + q = -10.404326023 Solving 2.5 + q = -10.404326023 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + q = -10.404326023 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + q = -10.404326023 + -2.5 q = -10.404326023 + -2.5 Combine like terms: -10.404326023 + -2.5 = -12.904326023 q = -12.904326023 Simplifying q = -12.904326023

Solution

The solution to the problem is based on the solutions from the subproblems. q = {7.904326023, -12.904326023}

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