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Simplifying 5t2 + 4t + -120 = 0 Reorder the terms: -120 + 4t + 5t2 = 0 Solving -120 + 4t + 5t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -24 + 0.8t + t2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + 0.8t + 24 + t2 = 0 + 24 Reorder the terms: -24 + 24 + 0.8t + t2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + 0.8t + t2 = 0 + 24 0.8t + t2 = 0 + 24 Combine like terms: 0 + 24 = 24 0.8t + t2 = 24 The t term is 0.8t. Take half its coefficient (0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. 0.8t + 0.16 + t2 = 24 + 0.16 Reorder the terms: 0.16 + 0.8t + t2 = 24 + 0.16 Combine like terms: 24 + 0.16 = 24.16 0.16 + 0.8t + t2 = 24.16 Factor a perfect square on the left side: (t + 0.4)(t + 0.4) = 24.16 Calculate the square root of the right side: 4.915282291 Break this problem into two subproblems by setting (t + 0.4) equal to 4.915282291 and -4.915282291.Subproblem 1
t + 0.4 = 4.915282291 Simplifying t + 0.4 = 4.915282291 Reorder the terms: 0.4 + t = 4.915282291 Solving 0.4 + t = 4.915282291 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + t = 4.915282291 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + t = 4.915282291 + -0.4 t = 4.915282291 + -0.4 Combine like terms: 4.915282291 + -0.4 = 4.515282291 t = 4.515282291 Simplifying t = 4.515282291Subproblem 2
t + 0.4 = -4.915282291 Simplifying t + 0.4 = -4.915282291 Reorder the terms: 0.4 + t = -4.915282291 Solving 0.4 + t = -4.915282291 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + t = -4.915282291 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + t = -4.915282291 + -0.4 t = -4.915282291 + -0.4 Combine like terms: -4.915282291 + -0.4 = -5.315282291 t = -5.315282291 Simplifying t = -5.315282291Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.515282291, -5.315282291}
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