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Simplifying (t + 3)(t + -4) = 5 Reorder the terms: (3 + t)(t + -4) = 5 Reorder the terms: (3 + t)(-4 + t) = 5 Multiply (3 + t) * (-4 + t) (3(-4 + t) + t(-4 + t)) = 5 ((-4 * 3 + t * 3) + t(-4 + t)) = 5 ((-12 + 3t) + t(-4 + t)) = 5 (-12 + 3t + (-4 * t + t * t)) = 5 (-12 + 3t + (-4t + t2)) = 5 Combine like terms: 3t + -4t = -1t (-12 + -1t + t2) = 5 Solving -12 + -1t + t2 = 5 Solving for variable 't'. Reorder the terms: -12 + -5 + -1t + t2 = 5 + -5 Combine like terms: -12 + -5 = -17 -17 + -1t + t2 = 5 + -5 Combine like terms: 5 + -5 = 0 -17 + -1t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '17' to each side of the equation. -17 + -1t + 17 + t2 = 0 + 17 Reorder the terms: -17 + 17 + -1t + t2 = 0 + 17 Combine like terms: -17 + 17 = 0 0 + -1t + t2 = 0 + 17 -1t + t2 = 0 + 17 Combine like terms: 0 + 17 = 17 -1t + t2 = 17 The t term is -1t. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1t + 0.25 + t2 = 17 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 17 + 0.25 Combine like terms: 17 + 0.25 = 17.25 0.25 + -1t + t2 = 17.25 Factor a perfect square on the left side: (t + -0.5)(t + -0.5) = 17.25 Calculate the square root of the right side: 4.153311931 Break this problem into two subproblems by setting (t + -0.5) equal to 4.153311931 and -4.153311931.Subproblem 1
t + -0.5 = 4.153311931 Simplifying t + -0.5 = 4.153311931 Reorder the terms: -0.5 + t = 4.153311931 Solving -0.5 + t = 4.153311931 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = 4.153311931 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = 4.153311931 + 0.5 t = 4.153311931 + 0.5 Combine like terms: 4.153311931 + 0.5 = 4.653311931 t = 4.653311931 Simplifying t = 4.653311931Subproblem 2
t + -0.5 = -4.153311931 Simplifying t + -0.5 = -4.153311931 Reorder the terms: -0.5 + t = -4.153311931 Solving -0.5 + t = -4.153311931 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = -4.153311931 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = -4.153311931 + 0.5 t = -4.153311931 + 0.5 Combine like terms: -4.153311931 + 0.5 = -3.653311931 t = -3.653311931 Simplifying t = -3.653311931Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.653311931, -3.653311931}
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