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Simplifying 0 = -16t2 + 410t + 84 Reorder the terms: 0 = 84 + 410t + -16t2 Solving 0 = 84 + 410t + -16t2 Solving for variable 't'. Combine like terms: 0 + -84 = -84 -84 + -410t + 16t2 = 84 + 410t + -16t2 + -84 + -410t + 16t2 Reorder the terms: -84 + -410t + 16t2 = 84 + -84 + 410t + -410t + -16t2 + 16t2 Combine like terms: 84 + -84 = 0 -84 + -410t + 16t2 = 0 + 410t + -410t + -16t2 + 16t2 -84 + -410t + 16t2 = 410t + -410t + -16t2 + 16t2 Combine like terms: 410t + -410t = 0 -84 + -410t + 16t2 = 0 + -16t2 + 16t2 -84 + -410t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -84 + -410t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-42 + -205t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-42 + -205t + 8t2)' equal to zero and attempt to solve: Simplifying -42 + -205t + 8t2 = 0 Solving -42 + -205t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -5.25 + -25.625t + t2 = 0 Move the constant term to the right: Add '5.25' to each side of the equation. -5.25 + -25.625t + 5.25 + t2 = 0 + 5.25 Reorder the terms: -5.25 + 5.25 + -25.625t + t2 = 0 + 5.25 Combine like terms: -5.25 + 5.25 = 0.00 0.00 + -25.625t + t2 = 0 + 5.25 -25.625t + t2 = 0 + 5.25 Combine like terms: 0 + 5.25 = 5.25 -25.625t + t2 = 5.25 The t term is -25.625t. Take half its coefficient (-12.8125). Square it (164.1601563) and add it to both sides. Add '164.1601563' to each side of the equation. -25.625t + 164.1601563 + t2 = 5.25 + 164.1601563 Reorder the terms: 164.1601563 + -25.625t + t2 = 5.25 + 164.1601563 Combine like terms: 5.25 + 164.1601563 = 169.4101563 164.1601563 + -25.625t + t2 = 169.4101563 Factor a perfect square on the left side: (t + -12.8125)(t + -12.8125) = 169.4101563 Calculate the square root of the right side: 13.015765682 Break this problem into two subproblems by setting (t + -12.8125) equal to 13.015765682 and -13.015765682.Subproblem 1
t + -12.8125 = 13.015765682 Simplifying t + -12.8125 = 13.015765682 Reorder the terms: -12.8125 + t = 13.015765682 Solving -12.8125 + t = 13.015765682 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '12.8125' to each side of the equation. -12.8125 + 12.8125 + t = 13.015765682 + 12.8125 Combine like terms: -12.8125 + 12.8125 = 0.0000 0.0000 + t = 13.015765682 + 12.8125 t = 13.015765682 + 12.8125 Combine like terms: 13.015765682 + 12.8125 = 25.828265682 t = 25.828265682 Simplifying t = 25.828265682Subproblem 2
t + -12.8125 = -13.015765682 Simplifying t + -12.8125 = -13.015765682 Reorder the terms: -12.8125 + t = -13.015765682 Solving -12.8125 + t = -13.015765682 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '12.8125' to each side of the equation. -12.8125 + 12.8125 + t = -13.015765682 + 12.8125 Combine like terms: -12.8125 + 12.8125 = 0.0000 0.0000 + t = -13.015765682 + 12.8125 t = -13.015765682 + 12.8125 Combine like terms: -13.015765682 + 12.8125 = -0.203265682 t = -0.203265682 Simplifying t = -0.203265682Solution
The solution to the problem is based on the solutions from the subproblems. t = {25.828265682, -0.203265682}Solution
t = {25.828265682, -0.203265682}
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