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Simplifying 5t2 + -7.5t + -30 = 0 Reorder the terms: -30 + -7.5t + 5t2 = 0 Solving -30 + -7.5t + 5t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -6 + -1.5t + t2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -1.5t + 6 + t2 = 0 + 6 Reorder the terms: -6 + 6 + -1.5t + t2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -1.5t + t2 = 0 + 6 -1.5t + t2 = 0 + 6 Combine like terms: 0 + 6 = 6 -1.5t + t2 = 6 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 6 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 6 + 0.5625 Combine like terms: 6 + 0.5625 = 6.5625 0.5625 + -1.5t + t2 = 6.5625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 6.5625 Calculate the square root of the right side: 2.561737691 Break this problem into two subproblems by setting (t + -0.75) equal to 2.561737691 and -2.561737691.Subproblem 1
t + -0.75 = 2.561737691 Simplifying t + -0.75 = 2.561737691 Reorder the terms: -0.75 + t = 2.561737691 Solving -0.75 + t = 2.561737691 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 2.561737691 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 2.561737691 + 0.75 t = 2.561737691 + 0.75 Combine like terms: 2.561737691 + 0.75 = 3.311737691 t = 3.311737691 Simplifying t = 3.311737691Subproblem 2
t + -0.75 = -2.561737691 Simplifying t + -0.75 = -2.561737691 Reorder the terms: -0.75 + t = -2.561737691 Solving -0.75 + t = -2.561737691 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -2.561737691 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -2.561737691 + 0.75 t = -2.561737691 + 0.75 Combine like terms: -2.561737691 + 0.75 = -1.811737691 t = -1.811737691 Simplifying t = -1.811737691Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.311737691, -1.811737691}
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