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Simplifying 100 = t2 + 5t Reorder the terms: 100 = 5t + t2 Solving 100 = 5t + t2 Solving for variable 't'. Reorder the terms: 100 + -5t + -1t2 = 5t + -5t + t2 + -1t2 Combine like terms: 5t + -5t = 0 100 + -5t + -1t2 = 0 + t2 + -1t2 100 + -5t + -1t2 = t2 + -1t2 Combine like terms: t2 + -1t2 = 0 100 + -5t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -100 + 5t + t2 = 0 Move the constant term to the right: Add '100' to each side of the equation. -100 + 5t + 100 + t2 = 0 + 100 Reorder the terms: -100 + 100 + 5t + t2 = 0 + 100 Combine like terms: -100 + 100 = 0 0 + 5t + t2 = 0 + 100 5t + t2 = 0 + 100 Combine like terms: 0 + 100 = 100 5t + t2 = 100 The t term is 5t. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5t + 6.25 + t2 = 100 + 6.25 Reorder the terms: 6.25 + 5t + t2 = 100 + 6.25 Combine like terms: 100 + 6.25 = 106.25 6.25 + 5t + t2 = 106.25 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 106.25 Calculate the square root of the right side: 10.307764064 Break this problem into two subproblems by setting (t + 2.5) equal to 10.307764064 and -10.307764064.Subproblem 1
t + 2.5 = 10.307764064 Simplifying t + 2.5 = 10.307764064 Reorder the terms: 2.5 + t = 10.307764064 Solving 2.5 + t = 10.307764064 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = 10.307764064 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 10.307764064 + -2.5 t = 10.307764064 + -2.5 Combine like terms: 10.307764064 + -2.5 = 7.807764064 t = 7.807764064 Simplifying t = 7.807764064Subproblem 2
t + 2.5 = -10.307764064 Simplifying t + 2.5 = -10.307764064 Reorder the terms: 2.5 + t = -10.307764064 Solving 2.5 + t = -10.307764064 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = -10.307764064 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -10.307764064 + -2.5 t = -10.307764064 + -2.5 Combine like terms: -10.307764064 + -2.5 = -12.807764064 t = -12.807764064 Simplifying t = -12.807764064Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.807764064, -12.807764064}
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