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Simplifying 8(t) = 0.3t2 + 0.1t + 4.2 Reorder the terms: 8t = 4.2 + 0.1t + 0.3t2 Solving 8t = 4.2 + 0.1t + 0.3t2 Solving for variable 't'. Reorder the terms: -4.2 + 8t + -0.1t + -0.3t2 = 4.2 + 0.1t + 0.3t2 + -4.2 + -0.1t + -0.3t2 Combine like terms: 8t + -0.1t = 7.9t -4.2 + 7.9t + -0.3t2 = 4.2 + 0.1t + 0.3t2 + -4.2 + -0.1t + -0.3t2 Reorder the terms: -4.2 + 7.9t + -0.3t2 = 4.2 + -4.2 + 0.1t + -0.1t + 0.3t2 + -0.3t2 Combine like terms: 4.2 + -4.2 = 0.0 -4.2 + 7.9t + -0.3t2 = 0.0 + 0.1t + -0.1t + 0.3t2 + -0.3t2 -4.2 + 7.9t + -0.3t2 = 0.1t + -0.1t + 0.3t2 + -0.3t2 Combine like terms: 0.1t + -0.1t = 0.0 -4.2 + 7.9t + -0.3t2 = 0.0 + 0.3t2 + -0.3t2 -4.2 + 7.9t + -0.3t2 = 0.3t2 + -0.3t2 Combine like terms: 0.3t2 + -0.3t2 = 0.0 -4.2 + 7.9t + -0.3t2 = 0.0 Begin completing the square. Divide all terms by -0.3 the coefficient of the squared term: Divide each side by '-0.3'. 14 + -26.33333333t + t2 = 0 Move the constant term to the right: Add '-14' to each side of the equation. 14 + -26.33333333t + -14 + t2 = 0 + -14 Reorder the terms: 14 + -14 + -26.33333333t + t2 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + -26.33333333t + t2 = 0 + -14 -26.33333333t + t2 = 0 + -14 Combine like terms: 0 + -14 = -14 -26.33333333t + t2 = -14 The t term is -26.33333333t. Take half its coefficient (-13.16666667). Square it (173.3611112) and add it to both sides. Add '173.3611112' to each side of the equation. -26.33333333t + 173.3611112 + t2 = -14 + 173.3611112 Reorder the terms: 173.3611112 + -26.33333333t + t2 = -14 + 173.3611112 Combine like terms: -14 + 173.3611112 = 159.3611112 173.3611112 + -26.33333333t + t2 = 159.3611112 Factor a perfect square on the left side: (t + -13.16666667)(t + -13.16666667) = 159.3611112 Calculate the square root of the right side: 12.623831083 Break this problem into two subproblems by setting (t + -13.16666667) equal to 12.623831083 and -12.623831083.Subproblem 1
t + -13.16666667 = 12.623831083 Simplifying t + -13.16666667 = 12.623831083 Reorder the terms: -13.16666667 + t = 12.623831083 Solving -13.16666667 + t = 12.623831083 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '13.16666667' to each side of the equation. -13.16666667 + 13.16666667 + t = 12.623831083 + 13.16666667 Combine like terms: -13.16666667 + 13.16666667 = 0.00000000 0.00000000 + t = 12.623831083 + 13.16666667 t = 12.623831083 + 13.16666667 Combine like terms: 12.623831083 + 13.16666667 = 25.790497753 t = 25.790497753 Simplifying t = 25.790497753Subproblem 2
t + -13.16666667 = -12.623831083 Simplifying t + -13.16666667 = -12.623831083 Reorder the terms: -13.16666667 + t = -12.623831083 Solving -13.16666667 + t = -12.623831083 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '13.16666667' to each side of the equation. -13.16666667 + 13.16666667 + t = -12.623831083 + 13.16666667 Combine like terms: -13.16666667 + 13.16666667 = 0.00000000 0.00000000 + t = -12.623831083 + 13.16666667 t = -12.623831083 + 13.16666667 Combine like terms: -12.623831083 + 13.16666667 = 0.542835587 t = 0.542835587 Simplifying t = 0.542835587Solution
The solution to the problem is based on the solutions from the subproblems. t = {25.790497753, 0.542835587}
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