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Simplifying 75(t) = 5t2 + 10t + 75 Reorder the terms: 75t = 75 + 10t + 5t2 Solving 75t = 75 + 10t + 5t2 Solving for variable 't'. Reorder the terms: -75 + 75t + -10t + -5t2 = 75 + 10t + 5t2 + -75 + -10t + -5t2 Combine like terms: 75t + -10t = 65t -75 + 65t + -5t2 = 75 + 10t + 5t2 + -75 + -10t + -5t2 Reorder the terms: -75 + 65t + -5t2 = 75 + -75 + 10t + -10t + 5t2 + -5t2 Combine like terms: 75 + -75 = 0 -75 + 65t + -5t2 = 0 + 10t + -10t + 5t2 + -5t2 -75 + 65t + -5t2 = 10t + -10t + 5t2 + -5t2 Combine like terms: 10t + -10t = 0 -75 + 65t + -5t2 = 0 + 5t2 + -5t2 -75 + 65t + -5t2 = 5t2 + -5t2 Combine like terms: 5t2 + -5t2 = 0 -75 + 65t + -5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-15 + 13t + -1t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-15 + 13t + -1t2)' equal to zero and attempt to solve: Simplifying -15 + 13t + -1t2 = 0 Solving -15 + 13t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 15 + -13t + t2 = 0 Move the constant term to the right: Add '-15' to each side of the equation. 15 + -13t + -15 + t2 = 0 + -15 Reorder the terms: 15 + -15 + -13t + t2 = 0 + -15 Combine like terms: 15 + -15 = 0 0 + -13t + t2 = 0 + -15 -13t + t2 = 0 + -15 Combine like terms: 0 + -15 = -15 -13t + t2 = -15 The t term is -13t. Take half its coefficient (-6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. -13t + 42.25 + t2 = -15 + 42.25 Reorder the terms: 42.25 + -13t + t2 = -15 + 42.25 Combine like terms: -15 + 42.25 = 27.25 42.25 + -13t + t2 = 27.25 Factor a perfect square on the left side: (t + -6.5)(t + -6.5) = 27.25 Calculate the square root of the right side: 5.220153254 Break this problem into two subproblems by setting (t + -6.5) equal to 5.220153254 and -5.220153254.Subproblem 1
t + -6.5 = 5.220153254 Simplifying t + -6.5 = 5.220153254 Reorder the terms: -6.5 + t = 5.220153254 Solving -6.5 + t = 5.220153254 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + t = 5.220153254 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + t = 5.220153254 + 6.5 t = 5.220153254 + 6.5 Combine like terms: 5.220153254 + 6.5 = 11.720153254 t = 11.720153254 Simplifying t = 11.720153254Subproblem 2
t + -6.5 = -5.220153254 Simplifying t + -6.5 = -5.220153254 Reorder the terms: -6.5 + t = -5.220153254 Solving -6.5 + t = -5.220153254 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + t = -5.220153254 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + t = -5.220153254 + 6.5 t = -5.220153254 + 6.5 Combine like terms: -5.220153254 + 6.5 = 1.279846746 t = 1.279846746 Simplifying t = 1.279846746Solution
The solution to the problem is based on the solutions from the subproblems. t = {11.720153254, 1.279846746}Solution
t = {11.720153254, 1.279846746}
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