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Simplifying 396 = (13 + 12w)(17 + 7w) Multiply (13 + 12w) * (17 + 7w) 396 = (13(17 + 7w) + 12w * (17 + 7w)) 396 = ((17 * 13 + 7w * 13) + 12w * (17 + 7w)) 396 = ((221 + 91w) + 12w * (17 + 7w)) 396 = (221 + 91w + (17 * 12w + 7w * 12w)) 396 = (221 + 91w + (204w + 84w2)) Combine like terms: 91w + 204w = 295w 396 = (221 + 295w + 84w2) Solving 396 = 221 + 295w + 84w2 Solving for variable 'w'. Combine like terms: 396 + -221 = 175 175 + -295w + -84w2 = 221 + 295w + 84w2 + -221 + -295w + -84w2 Reorder the terms: 175 + -295w + -84w2 = 221 + -221 + 295w + -295w + 84w2 + -84w2 Combine like terms: 221 + -221 = 0 175 + -295w + -84w2 = 0 + 295w + -295w + 84w2 + -84w2 175 + -295w + -84w2 = 295w + -295w + 84w2 + -84w2 Combine like terms: 295w + -295w = 0 175 + -295w + -84w2 = 0 + 84w2 + -84w2 175 + -295w + -84w2 = 84w2 + -84w2 Combine like terms: 84w2 + -84w2 = 0 175 + -295w + -84w2 = 0 Begin completing the square. Divide all terms by -84 the coefficient of the squared term: Divide each side by '-84'. -2.083333333 + 3.511904762w + w2 = 0 Move the constant term to the right: Add '2.083333333' to each side of the equation. -2.083333333 + 3.511904762w + 2.083333333 + w2 = 0 + 2.083333333 Reorder the terms: -2.083333333 + 2.083333333 + 3.511904762w + w2 = 0 + 2.083333333 Combine like terms: -2.083333333 + 2.083333333 = 0.000000000 0.000000000 + 3.511904762w + w2 = 0 + 2.083333333 3.511904762w + w2 = 0 + 2.083333333 Combine like terms: 0 + 2.083333333 = 2.083333333 3.511904762w + w2 = 2.083333333 The w term is 3.511904762w. Take half its coefficient (1.755952381). Square it (3.083368764) and add it to both sides. Add '3.083368764' to each side of the equation. 3.511904762w + 3.083368764 + w2 = 2.083333333 + 3.083368764 Reorder the terms: 3.083368764 + 3.511904762w + w2 = 2.083333333 + 3.083368764 Combine like terms: 2.083333333 + 3.083368764 = 5.166702097 3.083368764 + 3.511904762w + w2 = 5.166702097 Factor a perfect square on the left side: (w + 1.755952381)(w + 1.755952381) = 5.166702097 Calculate the square root of the right side: 2.273038076 Break this problem into two subproblems by setting (w + 1.755952381) equal to 2.273038076 and -2.273038076.Subproblem 1
w + 1.755952381 = 2.273038076 Simplifying w + 1.755952381 = 2.273038076 Reorder the terms: 1.755952381 + w = 2.273038076 Solving 1.755952381 + w = 2.273038076 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.755952381' to each side of the equation. 1.755952381 + -1.755952381 + w = 2.273038076 + -1.755952381 Combine like terms: 1.755952381 + -1.755952381 = 0.000000000 0.000000000 + w = 2.273038076 + -1.755952381 w = 2.273038076 + -1.755952381 Combine like terms: 2.273038076 + -1.755952381 = 0.517085695 w = 0.517085695 Simplifying w = 0.517085695Subproblem 2
w + 1.755952381 = -2.273038076 Simplifying w + 1.755952381 = -2.273038076 Reorder the terms: 1.755952381 + w = -2.273038076 Solving 1.755952381 + w = -2.273038076 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.755952381' to each side of the equation. 1.755952381 + -1.755952381 + w = -2.273038076 + -1.755952381 Combine like terms: 1.755952381 + -1.755952381 = 0.000000000 0.000000000 + w = -2.273038076 + -1.755952381 w = -2.273038076 + -1.755952381 Combine like terms: -2.273038076 + -1.755952381 = -4.028990457 w = -4.028990457 Simplifying w = -4.028990457Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.517085695, -4.028990457}
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