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Simplifying 15(t + 2) * 9t = 6 Reorder the terms: 15(2 + t) * 9t = 6 Reorder the terms for easier multiplication: 15 * 9t(2 + t) = 6 Multiply 15 * 9 135t(2 + t) = 6 (2 * 135t + t * 135t) = 6 (270t + 135t2) = 6 Solving 270t + 135t2 = 6 Solving for variable 't'. Reorder the terms: -6 + 270t + 135t2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + 270t + 135t2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-2 + 90t + 45t2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-2 + 90t + 45t2)' equal to zero and attempt to solve: Simplifying -2 + 90t + 45t2 = 0 Solving -2 + 90t + 45t2 = 0 Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. -0.04444444444 + 2t + t2 = 0 Move the constant term to the right: Add '0.04444444444' to each side of the equation. -0.04444444444 + 2t + 0.04444444444 + t2 = 0 + 0.04444444444 Reorder the terms: -0.04444444444 + 0.04444444444 + 2t + t2 = 0 + 0.04444444444 Combine like terms: -0.04444444444 + 0.04444444444 = 0.00000000000 0.00000000000 + 2t + t2 = 0 + 0.04444444444 2t + t2 = 0 + 0.04444444444 Combine like terms: 0 + 0.04444444444 = 0.04444444444 2t + t2 = 0.04444444444 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 = 0.04444444444 + 1 Reorder the terms: 1 + 2t + t2 = 0.04444444444 + 1 Combine like terms: 0.04444444444 + 1 = 1.04444444444 1 + 2t + t2 = 1.04444444444 Factor a perfect square on the left side: (t + 1)(t + 1) = 1.04444444444 Calculate the square root of the right side: 1.021980648 Break this problem into two subproblems by setting (t + 1) equal to 1.021980648 and -1.021980648.Subproblem 1
t + 1 = 1.021980648 Simplifying t + 1 = 1.021980648 Reorder the terms: 1 + t = 1.021980648 Solving 1 + t = 1.021980648 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 = 1.021980648 + -1 Combine like terms: 1 + -1 = 0 0 + t = 1.021980648 + -1 t = 1.021980648 + -1 Combine like terms: 1.021980648 + -1 = 0.021980648 t = 0.021980648 Simplifying t = 0.021980648Subproblem 2
t + 1 = -1.021980648 Simplifying t + 1 = -1.021980648 Reorder the terms: 1 + t = -1.021980648 Solving 1 + t = -1.021980648 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 = -1.021980648 + -1 Combine like terms: 1 + -1 = 0 0 + t = -1.021980648 + -1 t = -1.021980648 + -1 Combine like terms: -1.021980648 + -1 = -2.021980648 t = -2.021980648 Simplifying t = -2.021980648Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.021980648, -2.021980648}Solution
t = {0.021980648, -2.021980648}
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