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Simplifying 4t2 + 6t = 30 Reorder the terms: 6t + 4t2 = 30 Solving 6t + 4t2 = 30 Solving for variable 't'. Reorder the terms: -30 + 6t + 4t2 = 30 + -30 Combine like terms: 30 + -30 = 0 -30 + 6t + 4t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-15 + 3t + 2t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-15 + 3t + 2t2)' equal to zero and attempt to solve: Simplifying -15 + 3t + 2t2 = 0 Solving -15 + 3t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + 1.5t + t2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 1.5t + 7.5 + t2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 1.5t + t2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 1.5t + t2 = 0 + 7.5 1.5t + t2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 1.5t + t2 = 7.5 The t term is 1.5t. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5t + 0.5625 + t2 = 7.5 + 0.5625 Reorder the terms: 0.5625 + 1.5t + t2 = 7.5 + 0.5625 Combine like terms: 7.5 + 0.5625 = 8.0625 0.5625 + 1.5t + t2 = 8.0625 Factor a perfect square on the left side: (t + 0.75)(t + 0.75) = 8.0625 Calculate the square root of the right side: 2.839454173 Break this problem into two subproblems by setting (t + 0.75) equal to 2.839454173 and -2.839454173.Subproblem 1
t + 0.75 = 2.839454173 Simplifying t + 0.75 = 2.839454173 Reorder the terms: 0.75 + t = 2.839454173 Solving 0.75 + t = 2.839454173 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = 2.839454173 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = 2.839454173 + -0.75 t = 2.839454173 + -0.75 Combine like terms: 2.839454173 + -0.75 = 2.089454173 t = 2.089454173 Simplifying t = 2.089454173Subproblem 2
t + 0.75 = -2.839454173 Simplifying t + 0.75 = -2.839454173 Reorder the terms: 0.75 + t = -2.839454173 Solving 0.75 + t = -2.839454173 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = -2.839454173 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = -2.839454173 + -0.75 t = -2.839454173 + -0.75 Combine like terms: -2.839454173 + -0.75 = -3.589454173 t = -3.589454173 Simplifying t = -3.589454173Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.089454173, -3.589454173}Solution
t = {2.089454173, -3.589454173}
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