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Simplifying -10t2 + 9t + 90 = 0 Reorder the terms: 90 + 9t + -10t2 = 0 Solving 90 + 9t + -10t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -9 + -0.9t + t2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -0.9t + 9 + t2 = 0 + 9 Reorder the terms: -9 + 9 + -0.9t + t2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -0.9t + t2 = 0 + 9 -0.9t + t2 = 0 + 9 Combine like terms: 0 + 9 = 9 -0.9t + t2 = 9 The t term is -0.9t. Take half its coefficient (-0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. -0.9t + 0.2025 + t2 = 9 + 0.2025 Reorder the terms: 0.2025 + -0.9t + t2 = 9 + 0.2025 Combine like terms: 9 + 0.2025 = 9.2025 0.2025 + -0.9t + t2 = 9.2025 Factor a perfect square on the left side: (t + -0.45)(t + -0.45) = 9.2025 Calculate the square root of the right side: 3.033562262 Break this problem into two subproblems by setting (t + -0.45) equal to 3.033562262 and -3.033562262.Subproblem 1
t + -0.45 = 3.033562262 Simplifying t + -0.45 = 3.033562262 Reorder the terms: -0.45 + t = 3.033562262 Solving -0.45 + t = 3.033562262 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + t = 3.033562262 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + t = 3.033562262 + 0.45 t = 3.033562262 + 0.45 Combine like terms: 3.033562262 + 0.45 = 3.483562262 t = 3.483562262 Simplifying t = 3.483562262Subproblem 2
t + -0.45 = -3.033562262 Simplifying t + -0.45 = -3.033562262 Reorder the terms: -0.45 + t = -3.033562262 Solving -0.45 + t = -3.033562262 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + t = -3.033562262 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + t = -3.033562262 + 0.45 t = -3.033562262 + 0.45 Combine like terms: -3.033562262 + 0.45 = -2.583562262 t = -2.583562262 Simplifying t = -2.583562262Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.483562262, -2.583562262}
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