w^2+6w+15=53

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Solution for w^2+6w+15=53 equation:


Simplifying
w2 + 6w + 15 = 53

Reorder the terms:
15 + 6w + w2 = 53

Solving
15 + 6w + w2 = 53

Solving for variable 'w'.

Reorder the terms:
15 + -53 + 6w + w2 = 53 + -53

Combine like terms: 15 + -53 = -38
-38 + 6w + w2 = 53 + -53

Combine like terms: 53 + -53 = 0
-38 + 6w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '38' to each side of the equation.
-38 + 6w + 38 + w2 = 0 + 38

Reorder the terms:
-38 + 38 + 6w + w2 = 0 + 38

Combine like terms: -38 + 38 = 0
0 + 6w + w2 = 0 + 38
6w + w2 = 0 + 38

Combine like terms: 0 + 38 = 38
6w + w2 = 38

The w term is 6w.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6w + 9 + w2 = 38 + 9

Reorder the terms:
9 + 6w + w2 = 38 + 9

Combine like terms: 38 + 9 = 47
9 + 6w + w2 = 47

Factor a perfect square on the left side:
(w + 3)(w + 3) = 47

Calculate the square root of the right side: 6.8556546

Break this problem into two subproblems by setting 
(w + 3) equal to 6.8556546 and -6.8556546.

Subproblem 1

w + 3 = 6.8556546 Simplifying w + 3 = 6.8556546 Reorder the terms: 3 + w = 6.8556546 Solving 3 + w = 6.8556546 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + w = 6.8556546 + -3 Combine like terms: 3 + -3 = 0 0 + w = 6.8556546 + -3 w = 6.8556546 + -3 Combine like terms: 6.8556546 + -3 = 3.8556546 w = 3.8556546 Simplifying w = 3.8556546

Subproblem 2

w + 3 = -6.8556546 Simplifying w + 3 = -6.8556546 Reorder the terms: 3 + w = -6.8556546 Solving 3 + w = -6.8556546 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + w = -6.8556546 + -3 Combine like terms: 3 + -3 = 0 0 + w = -6.8556546 + -3 w = -6.8556546 + -3 Combine like terms: -6.8556546 + -3 = -9.8556546 w = -9.8556546 Simplifying w = -9.8556546

Solution

The solution to the problem is based on the solutions from the subproblems. w = {3.8556546, -9.8556546}

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