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Simplifying 4t2 = 5t + 1 Reorder the terms: 4t2 = 1 + 5t Solving 4t2 = 1 + 5t Solving for variable 't'. Reorder the terms: -1 + -5t + 4t2 = 1 + 5t + -1 + -5t Reorder the terms: -1 + -5t + 4t2 = 1 + -1 + 5t + -5t Combine like terms: 1 + -1 = 0 -1 + -5t + 4t2 = 0 + 5t + -5t -1 + -5t + 4t2 = 5t + -5t Combine like terms: 5t + -5t = 0 -1 + -5t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -1.25t + t2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1.25t + 0.25 + t2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1.25t + t2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1.25t + t2 = 0 + 0.25 -1.25t + t2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1.25t + t2 = 0.25 The t term is -1.25t. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25t + 0.390625 + t2 = 0.25 + 0.390625 Reorder the terms: 0.390625 + -1.25t + t2 = 0.25 + 0.390625 Combine like terms: 0.25 + 0.390625 = 0.640625 0.390625 + -1.25t + t2 = 0.640625 Factor a perfect square on the left side: (t + -0.625)(t + -0.625) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (t + -0.625) equal to 0.80039053 and -0.80039053.Subproblem 1
t + -0.625 = 0.80039053 Simplifying t + -0.625 = 0.80039053 Reorder the terms: -0.625 + t = 0.80039053 Solving -0.625 + t = 0.80039053 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + t = 0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = 0.80039053 + 0.625 t = 0.80039053 + 0.625 Combine like terms: 0.80039053 + 0.625 = 1.42539053 t = 1.42539053 Simplifying t = 1.42539053Subproblem 2
t + -0.625 = -0.80039053 Simplifying t + -0.625 = -0.80039053 Reorder the terms: -0.625 + t = -0.80039053 Solving -0.625 + t = -0.80039053 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + t = -0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = -0.80039053 + 0.625 t = -0.80039053 + 0.625 Combine like terms: -0.80039053 + 0.625 = -0.17539053 t = -0.17539053 Simplifying t = -0.17539053Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.42539053, -0.17539053}
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