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Simplifying q2 + 14q + 31 = 0 Reorder the terms: 31 + 14q + q2 = 0 Solving 31 + 14q + q2 = 0 Solving for variable 'q'. Begin completing the square. Move the constant term to the right: Add '-31' to each side of the equation. 31 + 14q + -31 + q2 = 0 + -31 Reorder the terms: 31 + -31 + 14q + q2 = 0 + -31 Combine like terms: 31 + -31 = 0 0 + 14q + q2 = 0 + -31 14q + q2 = 0 + -31 Combine like terms: 0 + -31 = -31 14q + q2 = -31 The q term is 14q. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14q + 49 + q2 = -31 + 49 Reorder the terms: 49 + 14q + q2 = -31 + 49 Combine like terms: -31 + 49 = 18 49 + 14q + q2 = 18 Factor a perfect square on the left side: (q + 7)(q + 7) = 18 Calculate the square root of the right side: 4.242640687 Break this problem into two subproblems by setting (q + 7) equal to 4.242640687 and -4.242640687.Subproblem 1
q + 7 = 4.242640687 Simplifying q + 7 = 4.242640687 Reorder the terms: 7 + q = 4.242640687 Solving 7 + q = 4.242640687 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + q = 4.242640687 + -7 Combine like terms: 7 + -7 = 0 0 + q = 4.242640687 + -7 q = 4.242640687 + -7 Combine like terms: 4.242640687 + -7 = -2.757359313 q = -2.757359313 Simplifying q = -2.757359313Subproblem 2
q + 7 = -4.242640687 Simplifying q + 7 = -4.242640687 Reorder the terms: 7 + q = -4.242640687 Solving 7 + q = -4.242640687 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + q = -4.242640687 + -7 Combine like terms: 7 + -7 = 0 0 + q = -4.242640687 + -7 q = -4.242640687 + -7 Combine like terms: -4.242640687 + -7 = -11.242640687 q = -11.242640687 Simplifying q = -11.242640687Solution
The solution to the problem is based on the solutions from the subproblems. q = {-2.757359313, -11.242640687}
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