7s+2=4s(s+1)

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Solution for 7s+2=4s(s+1) equation:


Simplifying
7s + 2 = 4s(s + 1)

Reorder the terms:
2 + 7s = 4s(s + 1)

Reorder the terms:
2 + 7s = 4s(1 + s)
2 + 7s = (1 * 4s + s * 4s)
2 + 7s = (4s + 4s2)

Solving
2 + 7s = 4s + 4s2

Solving for variable 's'.

Combine like terms: 7s + -4s = 3s
2 + 3s + -4s2 = 4s + 4s2 + -4s + -4s2

Reorder the terms:
2 + 3s + -4s2 = 4s + -4s + 4s2 + -4s2

Combine like terms: 4s + -4s = 0
2 + 3s + -4s2 = 0 + 4s2 + -4s2
2 + 3s + -4s2 = 4s2 + -4s2

Combine like terms: 4s2 + -4s2 = 0
2 + 3s + -4s2 = 0

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

Divide each side by '-4'.
-0.5 + -0.75s + s2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.75s + 0.5 + s2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.75s + s2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.75s + s2 = 0 + 0.5
-0.75s + s2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.75s + s2 = 0.5

The s term is -0.75s.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75s + 0.140625 + s2 = 0.5 + 0.140625

Reorder the terms:
0.140625 + -0.75s + s2 = 0.5 + 0.140625

Combine like terms: 0.5 + 0.140625 = 0.640625
0.140625 + -0.75s + s2 = 0.640625

Factor a perfect square on the left side:
(s + -0.375)(s + -0.375) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(s + -0.375) equal to 0.80039053 and -0.80039053.

Subproblem 1

s + -0.375 = 0.80039053 Simplifying s + -0.375 = 0.80039053 Reorder the terms: -0.375 + s = 0.80039053 Solving -0.375 + s = 0.80039053 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + s = 0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + s = 0.80039053 + 0.375 s = 0.80039053 + 0.375 Combine like terms: 0.80039053 + 0.375 = 1.17539053 s = 1.17539053 Simplifying s = 1.17539053

Subproblem 2

s + -0.375 = -0.80039053 Simplifying s + -0.375 = -0.80039053 Reorder the terms: -0.375 + s = -0.80039053 Solving -0.375 + s = -0.80039053 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + s = -0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + s = -0.80039053 + 0.375 s = -0.80039053 + 0.375 Combine like terms: -0.80039053 + 0.375 = -0.42539053 s = -0.42539053 Simplifying s = -0.42539053

Solution

The solution to the problem is based on the solutions from the subproblems. s = {1.17539053, -0.42539053}

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