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Simplifying -15(t2 + 135t + -270) = 0 Reorder the terms: -15(-270 + 135t + t2) = 0 (-270 * -15 + 135t * -15 + t2 * -15) = 0 (4050 + -2025t + -15t2) = 0 Solving 4050 + -2025t + -15t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '15'. 15(270 + -135t + -1t2) = 0 Ignore the factor 15.Subproblem 1
Set the factor '(270 + -135t + -1t2)' equal to zero and attempt to solve: Simplifying 270 + -135t + -1t2 = 0 Solving 270 + -135t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -270 + 135t + t2 = 0 Move the constant term to the right: Add '270' to each side of the equation. -270 + 135t + 270 + t2 = 0 + 270 Reorder the terms: -270 + 270 + 135t + t2 = 0 + 270 Combine like terms: -270 + 270 = 0 0 + 135t + t2 = 0 + 270 135t + t2 = 0 + 270 Combine like terms: 0 + 270 = 270 135t + t2 = 270 The t term is 135t. Take half its coefficient (67.5). Square it (4556.25) and add it to both sides. Add '4556.25' to each side of the equation. 135t + 4556.25 + t2 = 270 + 4556.25 Reorder the terms: 4556.25 + 135t + t2 = 270 + 4556.25 Combine like terms: 270 + 4556.25 = 4826.25 4556.25 + 135t + t2 = 4826.25 Factor a perfect square on the left side: (t + 67.5)(t + 67.5) = 4826.25 Calculate the square root of the right side: 69.471217061 Break this problem into two subproblems by setting (t + 67.5) equal to 69.471217061 and -69.471217061.Subproblem 1
t + 67.5 = 69.471217061 Simplifying t + 67.5 = 69.471217061 Reorder the terms: 67.5 + t = 69.471217061 Solving 67.5 + t = 69.471217061 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-67.5' to each side of the equation. 67.5 + -67.5 + t = 69.471217061 + -67.5 Combine like terms: 67.5 + -67.5 = 0.0 0.0 + t = 69.471217061 + -67.5 t = 69.471217061 + -67.5 Combine like terms: 69.471217061 + -67.5 = 1.971217061 t = 1.971217061 Simplifying t = 1.971217061Subproblem 2
t + 67.5 = -69.471217061 Simplifying t + 67.5 = -69.471217061 Reorder the terms: 67.5 + t = -69.471217061 Solving 67.5 + t = -69.471217061 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-67.5' to each side of the equation. 67.5 + -67.5 + t = -69.471217061 + -67.5 Combine like terms: 67.5 + -67.5 = 0.0 0.0 + t = -69.471217061 + -67.5 t = -69.471217061 + -67.5 Combine like terms: -69.471217061 + -67.5 = -136.971217061 t = -136.971217061 Simplifying t = -136.971217061Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.971217061, -136.971217061}Solution
t = {1.971217061, -136.971217061}
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