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Simplifying -4t2 + 30t + -12 = 0 Reorder the terms: -12 + 30t + -4t2 = 0 Solving -12 + 30t + -4t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(-6 + 15t + -2t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-6 + 15t + -2t2)' equal to zero and attempt to solve: Simplifying -6 + 15t + -2t2 = 0 Solving -6 + 15t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 3 + -7.5t + t2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -7.5t + -3 + t2 = 0 + -3 Reorder the terms: 3 + -3 + -7.5t + t2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -7.5t + t2 = 0 + -3 -7.5t + t2 = 0 + -3 Combine like terms: 0 + -3 = -3 -7.5t + t2 = -3 The t term is -7.5t. Take half its coefficient (-3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. -7.5t + 14.0625 + t2 = -3 + 14.0625 Reorder the terms: 14.0625 + -7.5t + t2 = -3 + 14.0625 Combine like terms: -3 + 14.0625 = 11.0625 14.0625 + -7.5t + t2 = 11.0625 Factor a perfect square on the left side: (t + -3.75)(t + -3.75) = 11.0625 Calculate the square root of the right side: 3.326033674 Break this problem into two subproblems by setting (t + -3.75) equal to 3.326033674 and -3.326033674.Subproblem 1
t + -3.75 = 3.326033674 Simplifying t + -3.75 = 3.326033674 Reorder the terms: -3.75 + t = 3.326033674 Solving -3.75 + t = 3.326033674 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + t = 3.326033674 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = 3.326033674 + 3.75 t = 3.326033674 + 3.75 Combine like terms: 3.326033674 + 3.75 = 7.076033674 t = 7.076033674 Simplifying t = 7.076033674Subproblem 2
t + -3.75 = -3.326033674 Simplifying t + -3.75 = -3.326033674 Reorder the terms: -3.75 + t = -3.326033674 Solving -3.75 + t = -3.326033674 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + t = -3.326033674 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = -3.326033674 + 3.75 t = -3.326033674 + 3.75 Combine like terms: -3.326033674 + 3.75 = 0.423966326 t = 0.423966326 Simplifying t = 0.423966326Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.076033674, 0.423966326}Solution
t = {7.076033674, 0.423966326}
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