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Simplifying 0 = 12h2 + -76h + 90 Reorder the terms: 0 = 90 + -76h + 12h2 Solving 0 = 90 + -76h + 12h2 Solving for variable 'h'. Combine like terms: 0 + -90 = -90 -90 + 76h + -12h2 = 90 + -76h + 12h2 + -90 + 76h + -12h2 Reorder the terms: -90 + 76h + -12h2 = 90 + -90 + -76h + 76h + 12h2 + -12h2 Combine like terms: 90 + -90 = 0 -90 + 76h + -12h2 = 0 + -76h + 76h + 12h2 + -12h2 -90 + 76h + -12h2 = -76h + 76h + 12h2 + -12h2 Combine like terms: -76h + 76h = 0 -90 + 76h + -12h2 = 0 + 12h2 + -12h2 -90 + 76h + -12h2 = 12h2 + -12h2 Combine like terms: 12h2 + -12h2 = 0 -90 + 76h + -12h2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-45 + 38h + -6h2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-45 + 38h + -6h2)' equal to zero and attempt to solve: Simplifying -45 + 38h + -6h2 = 0 Solving -45 + 38h + -6h2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. 7.5 + -6.333333333h + h2 = 0 Move the constant term to the right: Add '-7.5' to each side of the equation. 7.5 + -6.333333333h + -7.5 + h2 = 0 + -7.5 Reorder the terms: 7.5 + -7.5 + -6.333333333h + h2 = 0 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + -6.333333333h + h2 = 0 + -7.5 -6.333333333h + h2 = 0 + -7.5 Combine like terms: 0 + -7.5 = -7.5 -6.333333333h + h2 = -7.5 The h term is -6.333333333h. Take half its coefficient (-3.166666667). Square it (10.02777778) and add it to both sides. Add '10.02777778' to each side of the equation. -6.333333333h + 10.02777778 + h2 = -7.5 + 10.02777778 Reorder the terms: 10.02777778 + -6.333333333h + h2 = -7.5 + 10.02777778 Combine like terms: -7.5 + 10.02777778 = 2.52777778 10.02777778 + -6.333333333h + h2 = 2.52777778 Factor a perfect square on the left side: (h + -3.166666667)(h + -3.166666667) = 2.52777778 Calculate the square root of the right side: 1.58989867 Break this problem into two subproblems by setting (h + -3.166666667) equal to 1.58989867 and -1.58989867.Subproblem 1
h + -3.166666667 = 1.58989867 Simplifying h + -3.166666667 = 1.58989867 Reorder the terms: -3.166666667 + h = 1.58989867 Solving -3.166666667 + h = 1.58989867 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + h = 1.58989867 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + h = 1.58989867 + 3.166666667 h = 1.58989867 + 3.166666667 Combine like terms: 1.58989867 + 3.166666667 = 4.756565337 h = 4.756565337 Simplifying h = 4.756565337Subproblem 2
h + -3.166666667 = -1.58989867 Simplifying h + -3.166666667 = -1.58989867 Reorder the terms: -3.166666667 + h = -1.58989867 Solving -3.166666667 + h = -1.58989867 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + h = -1.58989867 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + h = -1.58989867 + 3.166666667 h = -1.58989867 + 3.166666667 Combine like terms: -1.58989867 + 3.166666667 = 1.576767997 h = 1.576767997 Simplifying h = 1.576767997Solution
The solution to the problem is based on the solutions from the subproblems. h = {4.756565337, 1.576767997}Solution
h = {4.756565337, 1.576767997}
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