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Simplifying 20 + -16t(16t) + 32t = t Remove parenthesis around (16t) 20 + -16t * 16t + 32t = t Reorder the terms for easier multiplication: 20 + -16 * 16t * t + 32t = t Multiply -16 * 16 20 + -256t * t + 32t = t Multiply t * t 20 + -256t2 + 32t = t Reorder the terms: 20 + 32t + -256t2 = t Solving 20 + 32t + -256t2 = t Solving for variable 't'. Reorder the terms: 20 + 32t + -1t + -256t2 = t + -1t Combine like terms: 32t + -1t = 31t 20 + 31t + -256t2 = t + -1t Combine like terms: t + -1t = 0 20 + 31t + -256t2 = 0 Begin completing the square. Divide all terms by -256 the coefficient of the squared term: Divide each side by '-256'. -0.078125 + -0.12109375t + t2 = 0 Move the constant term to the right: Add '0.078125' to each side of the equation. -0.078125 + -0.12109375t + 0.078125 + t2 = 0 + 0.078125 Reorder the terms: -0.078125 + 0.078125 + -0.12109375t + t2 = 0 + 0.078125 Combine like terms: -0.078125 + 0.078125 = 0.000000 0.000000 + -0.12109375t + t2 = 0 + 0.078125 -0.12109375t + t2 = 0 + 0.078125 Combine like terms: 0 + 0.078125 = 0.078125 -0.12109375t + t2 = 0.078125 The t term is -0.12109375t. Take half its coefficient (-0.060546875). Square it (0.003665924072) and add it to both sides. Add '0.003665924072' to each side of the equation. -0.12109375t + 0.003665924072 + t2 = 0.078125 + 0.003665924072 Reorder the terms: 0.003665924072 + -0.12109375t + t2 = 0.078125 + 0.003665924072 Combine like terms: 0.078125 + 0.003665924072 = 0.081790924072 0.003665924072 + -0.12109375t + t2 = 0.081790924072 Factor a perfect square on the left side: (t + -0.060546875)(t + -0.060546875) = 0.081790924072 Calculate the square root of the right side: 0.285991126 Break this problem into two subproblems by setting (t + -0.060546875) equal to 0.285991126 and -0.285991126.Subproblem 1
t + -0.060546875 = 0.285991126 Simplifying t + -0.060546875 = 0.285991126 Reorder the terms: -0.060546875 + t = 0.285991126 Solving -0.060546875 + t = 0.285991126 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.060546875' to each side of the equation. -0.060546875 + 0.060546875 + t = 0.285991126 + 0.060546875 Combine like terms: -0.060546875 + 0.060546875 = 0.000000000 0.000000000 + t = 0.285991126 + 0.060546875 t = 0.285991126 + 0.060546875 Combine like terms: 0.285991126 + 0.060546875 = 0.346538001 t = 0.346538001 Simplifying t = 0.346538001Subproblem 2
t + -0.060546875 = -0.285991126 Simplifying t + -0.060546875 = -0.285991126 Reorder the terms: -0.060546875 + t = -0.285991126 Solving -0.060546875 + t = -0.285991126 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.060546875' to each side of the equation. -0.060546875 + 0.060546875 + t = -0.285991126 + 0.060546875 Combine like terms: -0.060546875 + 0.060546875 = 0.000000000 0.000000000 + t = -0.285991126 + 0.060546875 t = -0.285991126 + 0.060546875 Combine like terms: -0.285991126 + 0.060546875 = -0.225444251 t = -0.225444251 Simplifying t = -0.225444251Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.346538001, -0.225444251}
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