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Simplifying 0 = 4(3t2 + -6t + -10) Reorder the terms: 0 = 4(-10 + -6t + 3t2) 0 = (-10 * 4 + -6t * 4 + 3t2 * 4) 0 = (-40 + -24t + 12t2) Solving 0 = -40 + -24t + 12t2 Solving for variable 't'. Combine like terms: 0 + 40 = 40 40 + 24t + -12t2 = -40 + -24t + 12t2 + 40 + 24t + -12t2 Reorder the terms: 40 + 24t + -12t2 = -40 + 40 + -24t + 24t + 12t2 + -12t2 Combine like terms: -40 + 40 = 0 40 + 24t + -12t2 = 0 + -24t + 24t + 12t2 + -12t2 40 + 24t + -12t2 = -24t + 24t + 12t2 + -12t2 Combine like terms: -24t + 24t = 0 40 + 24t + -12t2 = 0 + 12t2 + -12t2 40 + 24t + -12t2 = 12t2 + -12t2 Combine like terms: 12t2 + -12t2 = 0 40 + 24t + -12t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(10 + 6t + -3t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(10 + 6t + -3t2)' equal to zero and attempt to solve: Simplifying 10 + 6t + -3t2 = 0 Solving 10 + 6t + -3t2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -3.333333333 + -2t + t2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + -2t + 3.333333333 + t2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + -2t + t2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + -2t + t2 = 0 + 3.333333333 -2t + t2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 -2t + t2 = 3.333333333 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 = 3.333333333 + 1 Reorder the terms: 1 + -2t + t2 = 3.333333333 + 1 Combine like terms: 3.333333333 + 1 = 4.333333333 1 + -2t + t2 = 4.333333333 Factor a perfect square on the left side: (t + -1)(t + -1) = 4.333333333 Calculate the square root of the right side: 2.081665999 Break this problem into two subproblems by setting (t + -1) equal to 2.081665999 and -2.081665999.Subproblem 1
t + -1 = 2.081665999 Simplifying t + -1 = 2.081665999 Reorder the terms: -1 + t = 2.081665999 Solving -1 + t = 2.081665999 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 = 2.081665999 + 1 Combine like terms: -1 + 1 = 0 0 + t = 2.081665999 + 1 t = 2.081665999 + 1 Combine like terms: 2.081665999 + 1 = 3.081665999 t = 3.081665999 Simplifying t = 3.081665999Subproblem 2
t + -1 = -2.081665999 Simplifying t + -1 = -2.081665999 Reorder the terms: -1 + t = -2.081665999 Solving -1 + t = -2.081665999 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 = -2.081665999 + 1 Combine like terms: -1 + 1 = 0 0 + t = -2.081665999 + 1 t = -2.081665999 + 1 Combine like terms: -2.081665999 + 1 = -1.081665999 t = -1.081665999 Simplifying t = -1.081665999Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.081665999, -1.081665999}Solution
t = {3.081665999, -1.081665999}
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