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Simplifying 0 = 40t2 + 80t + -90 Reorder the terms: 0 = -90 + 80t + 40t2 Solving 0 = -90 + 80t + 40t2 Solving for variable 't'. Combine like terms: 0 + 90 = 90 90 + -80t + -40t2 = -90 + 80t + 40t2 + 90 + -80t + -40t2 Reorder the terms: 90 + -80t + -40t2 = -90 + 90 + 80t + -80t + 40t2 + -40t2 Combine like terms: -90 + 90 = 0 90 + -80t + -40t2 = 0 + 80t + -80t + 40t2 + -40t2 90 + -80t + -40t2 = 80t + -80t + 40t2 + -40t2 Combine like terms: 80t + -80t = 0 90 + -80t + -40t2 = 0 + 40t2 + -40t2 90 + -80t + -40t2 = 40t2 + -40t2 Combine like terms: 40t2 + -40t2 = 0 90 + -80t + -40t2 = 0 Factor out the Greatest Common Factor (GCF), '10'. 10(9 + -8t + -4t2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(9 + -8t + -4t2)' equal to zero and attempt to solve: Simplifying 9 + -8t + -4t2 = 0 Solving 9 + -8t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.25 + 2t + t2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + 2t + 2.25 + t2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + 2t + t2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + 2t + t2 = 0 + 2.25 2t + t2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 2t + t2 = 2.25 The t term is 2t. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2t + 1 + t2 = 2.25 + 1 Reorder the terms: 1 + 2t + t2 = 2.25 + 1 Combine like terms: 2.25 + 1 = 3.25 1 + 2t + t2 = 3.25 Factor a perfect square on the left side: (t + 1)(t + 1) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (t + 1) equal to 1.802775638 and -1.802775638.Subproblem 1
t + 1 = 1.802775638 Simplifying t + 1 = 1.802775638 Reorder the terms: 1 + t = 1.802775638 Solving 1 + t = 1.802775638 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = 1.802775638 + -1 Combine like terms: 1 + -1 = 0 0 + t = 1.802775638 + -1 t = 1.802775638 + -1 Combine like terms: 1.802775638 + -1 = 0.802775638 t = 0.802775638 Simplifying t = 0.802775638Subproblem 2
t + 1 = -1.802775638 Simplifying t + 1 = -1.802775638 Reorder the terms: 1 + t = -1.802775638 Solving 1 + t = -1.802775638 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = -1.802775638 + -1 Combine like terms: 1 + -1 = 0 0 + t = -1.802775638 + -1 t = -1.802775638 + -1 Combine like terms: -1.802775638 + -1 = -2.802775638 t = -2.802775638 Simplifying t = -2.802775638Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.802775638, -2.802775638}Solution
t = {0.802775638, -2.802775638}
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