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Simplifying 30 = 16t2 + 100t + 70 Reorder the terms: 30 = 70 + 100t + 16t2 Solving 30 = 70 + 100t + 16t2 Solving for variable 't'. Combine like terms: 30 + -70 = -40 -40 + -100t + -16t2 = 70 + 100t + 16t2 + -70 + -100t + -16t2 Reorder the terms: -40 + -100t + -16t2 = 70 + -70 + 100t + -100t + 16t2 + -16t2 Combine like terms: 70 + -70 = 0 -40 + -100t + -16t2 = 0 + 100t + -100t + 16t2 + -16t2 -40 + -100t + -16t2 = 100t + -100t + 16t2 + -16t2 Combine like terms: 100t + -100t = 0 -40 + -100t + -16t2 = 0 + 16t2 + -16t2 -40 + -100t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -40 + -100t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-4'. -4(10 + 25t + 4t2) = 0 Ignore the factor -4.Subproblem 1
Set the factor '(10 + 25t + 4t2)' equal to zero and attempt to solve: Simplifying 10 + 25t + 4t2 = 0 Solving 10 + 25t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2.5 + 6.25t + t2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 6.25t + -2.5 + t2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 6.25t + t2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 6.25t + t2 = 0 + -2.5 6.25t + t2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 6.25t + t2 = -2.5 The t term is 6.25t. Take half its coefficient (3.125). Square it (9.765625) and add it to both sides. Add '9.765625' to each side of the equation. 6.25t + 9.765625 + t2 = -2.5 + 9.765625 Reorder the terms: 9.765625 + 6.25t + t2 = -2.5 + 9.765625 Combine like terms: -2.5 + 9.765625 = 7.265625 9.765625 + 6.25t + t2 = 7.265625 Factor a perfect square on the left side: (t + 3.125)(t + 3.125) = 7.265625 Calculate the square root of the right side: 2.695482332 Break this problem into two subproblems by setting (t + 3.125) equal to 2.695482332 and -2.695482332.Subproblem 1
t + 3.125 = 2.695482332 Simplifying t + 3.125 = 2.695482332 Reorder the terms: 3.125 + t = 2.695482332 Solving 3.125 + t = 2.695482332 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.125' to each side of the equation. 3.125 + -3.125 + t = 2.695482332 + -3.125 Combine like terms: 3.125 + -3.125 = 0.000 0.000 + t = 2.695482332 + -3.125 t = 2.695482332 + -3.125 Combine like terms: 2.695482332 + -3.125 = -0.429517668 t = -0.429517668 Simplifying t = -0.429517668Subproblem 2
t + 3.125 = -2.695482332 Simplifying t + 3.125 = -2.695482332 Reorder the terms: 3.125 + t = -2.695482332 Solving 3.125 + t = -2.695482332 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.125' to each side of the equation. 3.125 + -3.125 + t = -2.695482332 + -3.125 Combine like terms: 3.125 + -3.125 = 0.000 0.000 + t = -2.695482332 + -3.125 t = -2.695482332 + -3.125 Combine like terms: -2.695482332 + -3.125 = -5.820482332 t = -5.820482332 Simplifying t = -5.820482332Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.429517668, -5.820482332}Solution
t = {-0.429517668, -5.820482332}
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