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Simplifying 6t2 + 10t + -18 = 0 Reorder the terms: -18 + 10t + 6t2 = 0 Solving -18 + 10t + 6t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(-9 + 5t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-9 + 5t + 3t2)' equal to zero and attempt to solve: Simplifying -9 + 5t + 3t2 = 0 Solving -9 + 5t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3 + 1.666666667t + t2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 1.666666667t + 3 + t2 = 0 + 3 Reorder the terms: -3 + 3 + 1.666666667t + t2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 1.666666667t + t2 = 0 + 3 1.666666667t + t2 = 0 + 3 Combine like terms: 0 + 3 = 3 1.666666667t + t2 = 3 The t term is 1.666666667t. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667t + 0.6944444447 + t2 = 3 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667t + t2 = 3 + 0.6944444447 Combine like terms: 3 + 0.6944444447 = 3.6944444447 0.6944444447 + 1.666666667t + t2 = 3.6944444447 Factor a perfect square on the left side: (t + 0.8333333335)(t + 0.8333333335) = 3.6944444447 Calculate the square root of the right side: 1.922093766 Break this problem into two subproblems by setting (t + 0.8333333335) equal to 1.922093766 and -1.922093766.Subproblem 1
t + 0.8333333335 = 1.922093766 Simplifying t + 0.8333333335 = 1.922093766 Reorder the terms: 0.8333333335 + t = 1.922093766 Solving 0.8333333335 + t = 1.922093766 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + t = 1.922093766 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + t = 1.922093766 + -0.8333333335 t = 1.922093766 + -0.8333333335 Combine like terms: 1.922093766 + -0.8333333335 = 1.0887604325 t = 1.0887604325 Simplifying t = 1.0887604325Subproblem 2
t + 0.8333333335 = -1.922093766 Simplifying t + 0.8333333335 = -1.922093766 Reorder the terms: 0.8333333335 + t = -1.922093766 Solving 0.8333333335 + t = -1.922093766 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + t = -1.922093766 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + t = -1.922093766 + -0.8333333335 t = -1.922093766 + -0.8333333335 Combine like terms: -1.922093766 + -0.8333333335 = -2.7554270995 t = -2.7554270995 Simplifying t = -2.7554270995Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.0887604325, -2.7554270995}Solution
t = {1.0887604325, -2.7554270995}
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