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Simplifying 10 = -16t2 + 20t + 12 Reorder the terms: 10 = 12 + 20t + -16t2 Solving 10 = 12 + 20t + -16t2 Solving for variable 't'. Combine like terms: 10 + -12 = -2 -2 + -20t + 16t2 = 12 + 20t + -16t2 + -12 + -20t + 16t2 Reorder the terms: -2 + -20t + 16t2 = 12 + -12 + 20t + -20t + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -2 + -20t + 16t2 = 0 + 20t + -20t + -16t2 + 16t2 -2 + -20t + 16t2 = 20t + -20t + -16t2 + 16t2 Combine like terms: 20t + -20t = 0 -2 + -20t + 16t2 = 0 + -16t2 + 16t2 -2 + -20t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -2 + -20t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + -10t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + -10t + 8t2)' equal to zero and attempt to solve: Simplifying -1 + -10t + 8t2 = 0 Solving -1 + -10t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + -1.25t + t2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -1.25t + 0.125 + t2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + -1.25t + t2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -1.25t + t2 = 0 + 0.125 -1.25t + t2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 -1.25t + t2 = 0.125 The t term is -1.25t. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25t + 0.390625 + t2 = 0.125 + 0.390625 Reorder the terms: 0.390625 + -1.25t + t2 = 0.125 + 0.390625 Combine like terms: 0.125 + 0.390625 = 0.515625 0.390625 + -1.25t + t2 = 0.515625 Factor a perfect square on the left side: (t + -0.625)(t + -0.625) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (t + -0.625) equal to 0.718070331 and -0.718070331.Subproblem 1
t + -0.625 = 0.718070331 Simplifying t + -0.625 = 0.718070331 Reorder the terms: -0.625 + t = 0.718070331 Solving -0.625 + t = 0.718070331 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + t = 0.718070331 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = 0.718070331 + 0.625 t = 0.718070331 + 0.625 Combine like terms: 0.718070331 + 0.625 = 1.343070331 t = 1.343070331 Simplifying t = 1.343070331Subproblem 2
t + -0.625 = -0.718070331 Simplifying t + -0.625 = -0.718070331 Reorder the terms: -0.625 + t = -0.718070331 Solving -0.625 + t = -0.718070331 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + t = -0.718070331 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = -0.718070331 + 0.625 t = -0.718070331 + 0.625 Combine like terms: -0.718070331 + 0.625 = -0.093070331 t = -0.093070331 Simplifying t = -0.093070331Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.343070331, -0.093070331}Solution
t = {1.343070331, -0.093070331}
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