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Simplifying -16t2 + 90t + 3 = 0 Reorder the terms: 3 + 90t + -16t2 = 0 Solving 3 + 90t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.1875 + -5.625t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -5.625t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -5.625t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -5.625t + t2 = 0 + 0.1875 -5.625t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -5.625t + t2 = 0.1875 The t term is -5.625t. Take half its coefficient (-2.8125). Square it (7.91015625) and add it to both sides. Add '7.91015625' to each side of the equation. -5.625t + 7.91015625 + t2 = 0.1875 + 7.91015625 Reorder the terms: 7.91015625 + -5.625t + t2 = 0.1875 + 7.91015625 Combine like terms: 0.1875 + 7.91015625 = 8.09765625 7.91015625 + -5.625t + t2 = 8.09765625 Factor a perfect square on the left side: (t + -2.8125)(t + -2.8125) = 8.09765625 Calculate the square root of the right side: 2.845638109 Break this problem into two subproblems by setting (t + -2.8125) equal to 2.845638109 and -2.845638109.Subproblem 1
t + -2.8125 = 2.845638109 Simplifying t + -2.8125 = 2.845638109 Reorder the terms: -2.8125 + t = 2.845638109 Solving -2.8125 + t = 2.845638109 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.8125' to each side of the equation. -2.8125 + 2.8125 + t = 2.845638109 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = 2.845638109 + 2.8125 t = 2.845638109 + 2.8125 Combine like terms: 2.845638109 + 2.8125 = 5.658138109 t = 5.658138109 Simplifying t = 5.658138109Subproblem 2
t + -2.8125 = -2.845638109 Simplifying t + -2.8125 = -2.845638109 Reorder the terms: -2.8125 + t = -2.845638109 Solving -2.8125 + t = -2.845638109 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.8125' to each side of the equation. -2.8125 + 2.8125 + t = -2.845638109 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = -2.845638109 + 2.8125 t = -2.845638109 + 2.8125 Combine like terms: -2.845638109 + 2.8125 = -0.033138109 t = -0.033138109 Simplifying t = -0.033138109Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.658138109, -0.033138109}
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