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Simplifying -16t2 + 28t + 32 = 0 Reorder the terms: 32 + 28t + -16t2 = 0 Solving 32 + 28t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(8 + 7t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(8 + 7t + -4t2)' equal to zero and attempt to solve: Simplifying 8 + 7t + -4t2 = 0 Solving 8 + 7t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2 + -1.75t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -1.75t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + -1.75t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -1.75t + t2 = 0 + 2 -1.75t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 -1.75t + t2 = 2 The t term is -1.75t. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75t + 0.765625 + t2 = 2 + 0.765625 Reorder the terms: 0.765625 + -1.75t + t2 = 2 + 0.765625 Combine like terms: 2 + 0.765625 = 2.765625 0.765625 + -1.75t + t2 = 2.765625 Factor a perfect square on the left side: (t + -0.875)(t + -0.875) = 2.765625 Calculate the square root of the right side: 1.663016837 Break this problem into two subproblems by setting (t + -0.875) equal to 1.663016837 and -1.663016837.Subproblem 1
t + -0.875 = 1.663016837 Simplifying t + -0.875 = 1.663016837 Reorder the terms: -0.875 + t = 1.663016837 Solving -0.875 + t = 1.663016837 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = 1.663016837 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = 1.663016837 + 0.875 t = 1.663016837 + 0.875 Combine like terms: 1.663016837 + 0.875 = 2.538016837 t = 2.538016837 Simplifying t = 2.538016837Subproblem 2
t + -0.875 = -1.663016837 Simplifying t + -0.875 = -1.663016837 Reorder the terms: -0.875 + t = -1.663016837 Solving -0.875 + t = -1.663016837 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = -1.663016837 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = -1.663016837 + 0.875 t = -1.663016837 + 0.875 Combine like terms: -1.663016837 + 0.875 = -0.788016837 t = -0.788016837 Simplifying t = -0.788016837Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.538016837, -0.788016837}Solution
t = {2.538016837, -0.788016837}
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