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Simplifying 50 = -5(t + -5)(t + 3) Reorder the terms: 50 = -5(-5 + t)(t + 3) Reorder the terms: 50 = -5(-5 + t)(3 + t) Multiply (-5 + t) * (3 + t) 50 = -5(-5(3 + t) + t(3 + t)) 50 = -5((3 * -5 + t * -5) + t(3 + t)) 50 = -5((-15 + -5t) + t(3 + t)) 50 = -5(-15 + -5t + (3 * t + t * t)) 50 = -5(-15 + -5t + (3t + t2)) Combine like terms: -5t + 3t = -2t 50 = -5(-15 + -2t + t2) 50 = (-15 * -5 + -2t * -5 + t2 * -5) 50 = (75 + 10t + -5t2) Solving 50 = 75 + 10t + -5t2 Solving for variable 't'. Combine like terms: 50 + -75 = -25 -25 + -10t + 5t2 = 75 + 10t + -5t2 + -75 + -10t + 5t2 Reorder the terms: -25 + -10t + 5t2 = 75 + -75 + 10t + -10t + -5t2 + 5t2 Combine like terms: 75 + -75 = 0 -25 + -10t + 5t2 = 0 + 10t + -10t + -5t2 + 5t2 -25 + -10t + 5t2 = 10t + -10t + -5t2 + 5t2 Combine like terms: 10t + -10t = 0 -25 + -10t + 5t2 = 0 + -5t2 + 5t2 -25 + -10t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 -25 + -10t + 5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-5 + -2t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-5 + -2t + t2)' equal to zero and attempt to solve: Simplifying -5 + -2t + t2 = 0 Solving -5 + -2t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + -2t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -2t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2t + t2 = 0 + 5 -2t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2t + t2 = 5 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 = 5 + 1 Reorder the terms: 1 + -2t + t2 = 5 + 1 Combine like terms: 5 + 1 = 6 1 + -2t + t2 = 6 Factor a perfect square on the left side: (t + -1)(t + -1) = 6 Calculate the square root of the right side: 2.449489743 Break this problem into two subproblems by setting (t + -1) equal to 2.449489743 and -2.449489743.Subproblem 1
t + -1 = 2.449489743 Simplifying t + -1 = 2.449489743 Reorder the terms: -1 + t = 2.449489743 Solving -1 + t = 2.449489743 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 = 2.449489743 + 1 Combine like terms: -1 + 1 = 0 0 + t = 2.449489743 + 1 t = 2.449489743 + 1 Combine like terms: 2.449489743 + 1 = 3.449489743 t = 3.449489743 Simplifying t = 3.449489743Subproblem 2
t + -1 = -2.449489743 Simplifying t + -1 = -2.449489743 Reorder the terms: -1 + t = -2.449489743 Solving -1 + t = -2.449489743 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 = -2.449489743 + 1 Combine like terms: -1 + 1 = 0 0 + t = -2.449489743 + 1 t = -2.449489743 + 1 Combine like terms: -2.449489743 + 1 = -1.449489743 t = -1.449489743 Simplifying t = -1.449489743Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.449489743, -1.449489743}Solution
t = {3.449489743, -1.449489743}
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