7q(3q+14)=18

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Solution for 7q(3q+14)=18 equation:


Simplifying
7q(3q + 14) = 18

Reorder the terms:
7q(14 + 3q) = 18
(14 * 7q + 3q * 7q) = 18
(98q + 21q2) = 18

Solving
98q + 21q2 = 18

Solving for variable 'q'.

Reorder the terms:
-18 + 98q + 21q2 = 18 + -18

Combine like terms: 18 + -18 = 0
-18 + 98q + 21q2 = 0

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

Divide each side by '21'.
-0.8571428571 + 4.666666667q + q2 = 0

Move the constant term to the right:

Add '0.8571428571' to each side of the equation.
-0.8571428571 + 4.666666667q + 0.8571428571 + q2 = 0 + 0.8571428571

Reorder the terms:
-0.8571428571 + 0.8571428571 + 4.666666667q + q2 = 0 + 0.8571428571

Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000
0.0000000000 + 4.666666667q + q2 = 0 + 0.8571428571
4.666666667q + q2 = 0 + 0.8571428571

Combine like terms: 0 + 0.8571428571 = 0.8571428571
4.666666667q + q2 = 0.8571428571

The q term is 4.666666667q.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667q + 5.444444448 + q2 = 0.8571428571 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667q + q2 = 0.8571428571 + 5.444444448

Combine like terms: 0.8571428571 + 5.444444448 = 6.3015873051
5.444444448 + 4.666666667q + q2 = 6.3015873051

Factor a perfect square on the left side:
(q + 2.333333334)(q + 2.333333334) = 6.3015873051

Calculate the square root of the right side: 2.510296258

Break this problem into two subproblems by setting 
(q + 2.333333334) equal to 2.510296258 and -2.510296258.

Subproblem 1

q + 2.333333334 = 2.510296258 Simplifying q + 2.333333334 = 2.510296258 Reorder the terms: 2.333333334 + q = 2.510296258 Solving 2.333333334 + q = 2.510296258 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + q = 2.510296258 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + q = 2.510296258 + -2.333333334 q = 2.510296258 + -2.333333334 Combine like terms: 2.510296258 + -2.333333334 = 0.176962924 q = 0.176962924 Simplifying q = 0.176962924

Subproblem 2

q + 2.333333334 = -2.510296258 Simplifying q + 2.333333334 = -2.510296258 Reorder the terms: 2.333333334 + q = -2.510296258 Solving 2.333333334 + q = -2.510296258 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + q = -2.510296258 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + q = -2.510296258 + -2.333333334 q = -2.510296258 + -2.333333334 Combine like terms: -2.510296258 + -2.333333334 = -4.843629592 q = -4.843629592 Simplifying q = -4.843629592

Solution

The solution to the problem is based on the solutions from the subproblems. q = {0.176962924, -4.843629592}

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