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Simplifying (2t + 1)(t + -0.75) = 0 Reorder the terms: (1 + 2t)(t + -0.75) = 0 Reorder the terms: (1 + 2t)(-0.75 + t) = 0 Multiply (1 + 2t) * (-0.75 + t) (1(-0.75 + t) + 2t * (-0.75 + t)) = 0 ((-0.75 * 1 + t * 1) + 2t * (-0.75 + t)) = 0 ((-0.75 + 1t) + 2t * (-0.75 + t)) = 0 (-0.75 + 1t + (-0.75 * 2t + t * 2t)) = 0 (-0.75 + 1t + (-1.5t + 2t2)) = 0 Combine like terms: 1t + -1.5t = -0.5t (-0.75 + -0.5t + 2t2) = 0 Solving -0.75 + -0.5t + 2t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.375 + -0.25t + t2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -0.25t + 0.375 + t2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -0.25t + t2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -0.25t + t2 = 0 + 0.375 -0.25t + t2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -0.25t + t2 = 0.375 The t term is -0.25t. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25t + 0.015625 + t2 = 0.375 + 0.015625 Reorder the terms: 0.015625 + -0.25t + t2 = 0.375 + 0.015625 Combine like terms: 0.375 + 0.015625 = 0.390625 0.015625 + -0.25t + t2 = 0.390625 Factor a perfect square on the left side: (t + -0.125)(t + -0.125) = 0.390625 Calculate the square root of the right side: 0.625 Break this problem into two subproblems by setting (t + -0.125) equal to 0.625 and -0.625.Subproblem 1
t + -0.125 = 0.625 Simplifying t + -0.125 = 0.625 Reorder the terms: -0.125 + t = 0.625 Solving -0.125 + t = 0.625 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + t = 0.625 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + t = 0.625 + 0.125 t = 0.625 + 0.125 Combine like terms: 0.625 + 0.125 = 0.75 t = 0.75 Simplifying t = 0.75Subproblem 2
t + -0.125 = -0.625 Simplifying t + -0.125 = -0.625 Reorder the terms: -0.125 + t = -0.625 Solving -0.125 + t = -0.625 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + t = -0.625 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + t = -0.625 + 0.125 t = -0.625 + 0.125 Combine like terms: -0.625 + 0.125 = -0.5 t = -0.5 Simplifying t = -0.5Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.75, -0.5}
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