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Simplifying 500 = 8t2 + 4t + 400 Reorder the terms: 500 = 400 + 4t + 8t2 Solving 500 = 400 + 4t + 8t2 Solving for variable 't'. Combine like terms: 500 + -400 = 100 100 + -4t + -8t2 = 400 + 4t + 8t2 + -400 + -4t + -8t2 Reorder the terms: 100 + -4t + -8t2 = 400 + -400 + 4t + -4t + 8t2 + -8t2 Combine like terms: 400 + -400 = 0 100 + -4t + -8t2 = 0 + 4t + -4t + 8t2 + -8t2 100 + -4t + -8t2 = 4t + -4t + 8t2 + -8t2 Combine like terms: 4t + -4t = 0 100 + -4t + -8t2 = 0 + 8t2 + -8t2 100 + -4t + -8t2 = 8t2 + -8t2 Combine like terms: 8t2 + -8t2 = 0 100 + -4t + -8t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(25 + -1t + -2t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(25 + -1t + -2t2)' equal to zero and attempt to solve: Simplifying 25 + -1t + -2t2 = 0 Solving 25 + -1t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -12.5 + 0.5t + t2 = 0 Move the constant term to the right: Add '12.5' to each side of the equation. -12.5 + 0.5t + 12.5 + t2 = 0 + 12.5 Reorder the terms: -12.5 + 12.5 + 0.5t + t2 = 0 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + 0.5t + t2 = 0 + 12.5 0.5t + t2 = 0 + 12.5 Combine like terms: 0 + 12.5 = 12.5 0.5t + t2 = 12.5 The t term is 0.5t. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5t + 0.0625 + t2 = 12.5 + 0.0625 Reorder the terms: 0.0625 + 0.5t + t2 = 12.5 + 0.0625 Combine like terms: 12.5 + 0.0625 = 12.5625 0.0625 + 0.5t + t2 = 12.5625 Factor a perfect square on the left side: (t + 0.25)(t + 0.25) = 12.5625 Calculate the square root of the right side: 3.54436172 Break this problem into two subproblems by setting (t + 0.25) equal to 3.54436172 and -3.54436172.Subproblem 1
t + 0.25 = 3.54436172 Simplifying t + 0.25 = 3.54436172 Reorder the terms: 0.25 + t = 3.54436172 Solving 0.25 + t = 3.54436172 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + t = 3.54436172 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + t = 3.54436172 + -0.25 t = 3.54436172 + -0.25 Combine like terms: 3.54436172 + -0.25 = 3.29436172 t = 3.29436172 Simplifying t = 3.29436172Subproblem 2
t + 0.25 = -3.54436172 Simplifying t + 0.25 = -3.54436172 Reorder the terms: 0.25 + t = -3.54436172 Solving 0.25 + t = -3.54436172 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + t = -3.54436172 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + t = -3.54436172 + -0.25 t = -3.54436172 + -0.25 Combine like terms: -3.54436172 + -0.25 = -3.79436172 t = -3.79436172 Simplifying t = -3.79436172Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.29436172, -3.79436172}Solution
t = {3.29436172, -3.79436172}
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