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Simplifying -4(4t2 + -19t + -27) = 0 Reorder the terms: -4(-27 + -19t + 4t2) = 0 (-27 * -4 + -19t * -4 + 4t2 * -4) = 0 (108 + 76t + -16t2) = 0 Solving 108 + 76t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(27 + 19t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(27 + 19t + -4t2)' equal to zero and attempt to solve: Simplifying 27 + 19t + -4t2 = 0 Solving 27 + 19t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -6.75 + -4.75t + t2 = 0 Move the constant term to the right: Add '6.75' to each side of the equation. -6.75 + -4.75t + 6.75 + t2 = 0 + 6.75 Reorder the terms: -6.75 + 6.75 + -4.75t + t2 = 0 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + -4.75t + t2 = 0 + 6.75 -4.75t + t2 = 0 + 6.75 Combine like terms: 0 + 6.75 = 6.75 -4.75t + t2 = 6.75 The t term is -4.75t. Take half its coefficient (-2.375). Square it (5.640625) and add it to both sides. Add '5.640625' to each side of the equation. -4.75t + 5.640625 + t2 = 6.75 + 5.640625 Reorder the terms: 5.640625 + -4.75t + t2 = 6.75 + 5.640625 Combine like terms: 6.75 + 5.640625 = 12.390625 5.640625 + -4.75t + t2 = 12.390625 Factor a perfect square on the left side: (t + -2.375)(t + -2.375) = 12.390625 Calculate the square root of the right side: 3.52003196 Break this problem into two subproblems by setting (t + -2.375) equal to 3.52003196 and -3.52003196.Subproblem 1
t + -2.375 = 3.52003196 Simplifying t + -2.375 = 3.52003196 Reorder the terms: -2.375 + t = 3.52003196 Solving -2.375 + t = 3.52003196 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + t = 3.52003196 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + t = 3.52003196 + 2.375 t = 3.52003196 + 2.375 Combine like terms: 3.52003196 + 2.375 = 5.89503196 t = 5.89503196 Simplifying t = 5.89503196Subproblem 2
t + -2.375 = -3.52003196 Simplifying t + -2.375 = -3.52003196 Reorder the terms: -2.375 + t = -3.52003196 Solving -2.375 + t = -3.52003196 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + t = -3.52003196 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + t = -3.52003196 + 2.375 t = -3.52003196 + 2.375 Combine like terms: -3.52003196 + 2.375 = -1.14503196 t = -1.14503196 Simplifying t = -1.14503196Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.89503196, -1.14503196}Solution
t = {5.89503196, -1.14503196}
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