7w^2+12w=8w+9

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Solution for 7w^2+12w=8w+9 equation:


Simplifying
7w2 + 12w = 8w + 9

Reorder the terms:
12w + 7w2 = 8w + 9

Reorder the terms:
12w + 7w2 = 9 + 8w

Solving
12w + 7w2 = 9 + 8w

Solving for variable 'w'.

Reorder the terms:
-9 + 12w + -8w + 7w2 = 9 + 8w + -9 + -8w

Combine like terms: 12w + -8w = 4w
-9 + 4w + 7w2 = 9 + 8w + -9 + -8w

Reorder the terms:
-9 + 4w + 7w2 = 9 + -9 + 8w + -8w

Combine like terms: 9 + -9 = 0
-9 + 4w + 7w2 = 0 + 8w + -8w
-9 + 4w + 7w2 = 8w + -8w

Combine like terms: 8w + -8w = 0
-9 + 4w + 7w2 = 0

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

Divide each side by '7'.
-1.285714286 + 0.5714285714w + w2 = 0

Move the constant term to the right:

Add '1.285714286' to each side of the equation.
-1.285714286 + 0.5714285714w + 1.285714286 + w2 = 0 + 1.285714286

Reorder the terms:
-1.285714286 + 1.285714286 + 0.5714285714w + w2 = 0 + 1.285714286

Combine like terms: -1.285714286 + 1.285714286 = 0.000000000
0.000000000 + 0.5714285714w + w2 = 0 + 1.285714286
0.5714285714w + w2 = 0 + 1.285714286

Combine like terms: 0 + 1.285714286 = 1.285714286
0.5714285714w + w2 = 1.285714286

The w term is 0.5714285714w.  Take half its coefficient (0.2857142857).
Square it (0.08163265305) and add it to both sides.

Add '0.08163265305' to each side of the equation.
0.5714285714w + 0.08163265305 + w2 = 1.285714286 + 0.08163265305

Reorder the terms:
0.08163265305 + 0.5714285714w + w2 = 1.285714286 + 0.08163265305

Combine like terms: 1.285714286 + 0.08163265305 = 1.36734693905
0.08163265305 + 0.5714285714w + w2 = 1.36734693905

Factor a perfect square on the left side:
(w + 0.2857142857)(w + 0.2857142857) = 1.36734693905

Calculate the square root of the right side: 1.16933611

Break this problem into two subproblems by setting 
(w + 0.2857142857) equal to 1.16933611 and -1.16933611.

Subproblem 1

w + 0.2857142857 = 1.16933611 Simplifying w + 0.2857142857 = 1.16933611 Reorder the terms: 0.2857142857 + w = 1.16933611 Solving 0.2857142857 + w = 1.16933611 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.2857142857' to each side of the equation. 0.2857142857 + -0.2857142857 + w = 1.16933611 + -0.2857142857 Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000 0.0000000000 + w = 1.16933611 + -0.2857142857 w = 1.16933611 + -0.2857142857 Combine like terms: 1.16933611 + -0.2857142857 = 0.8836218243 w = 0.8836218243 Simplifying w = 0.8836218243

Subproblem 2

w + 0.2857142857 = -1.16933611 Simplifying w + 0.2857142857 = -1.16933611 Reorder the terms: 0.2857142857 + w = -1.16933611 Solving 0.2857142857 + w = -1.16933611 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.2857142857' to each side of the equation. 0.2857142857 + -0.2857142857 + w = -1.16933611 + -0.2857142857 Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000 0.0000000000 + w = -1.16933611 + -0.2857142857 w = -1.16933611 + -0.2857142857 Combine like terms: -1.16933611 + -0.2857142857 = -1.4550503957 w = -1.4550503957 Simplifying w = -1.4550503957

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.8836218243, -1.4550503957}

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