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Simplifying 1400 = 2t2 + 5t + 7 Reorder the terms: 1400 = 7 + 5t + 2t2 Solving 1400 = 7 + 5t + 2t2 Solving for variable 't'. Combine like terms: 1400 + -7 = 1393 1393 + -5t + -2t2 = 7 + 5t + 2t2 + -7 + -5t + -2t2 Reorder the terms: 1393 + -5t + -2t2 = 7 + -7 + 5t + -5t + 2t2 + -2t2 Combine like terms: 7 + -7 = 0 1393 + -5t + -2t2 = 0 + 5t + -5t + 2t2 + -2t2 1393 + -5t + -2t2 = 5t + -5t + 2t2 + -2t2 Combine like terms: 5t + -5t = 0 1393 + -5t + -2t2 = 0 + 2t2 + -2t2 1393 + -5t + -2t2 = 2t2 + -2t2 Combine like terms: 2t2 + -2t2 = 0 1393 + -5t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -696.5 + 2.5t + t2 = 0 Move the constant term to the right: Add '696.5' to each side of the equation. -696.5 + 2.5t + 696.5 + t2 = 0 + 696.5 Reorder the terms: -696.5 + 696.5 + 2.5t + t2 = 0 + 696.5 Combine like terms: -696.5 + 696.5 = 0.0 0.0 + 2.5t + t2 = 0 + 696.5 2.5t + t2 = 0 + 696.5 Combine like terms: 0 + 696.5 = 696.5 2.5t + t2 = 696.5 The t term is 2.5t. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5t + 1.5625 + t2 = 696.5 + 1.5625 Reorder the terms: 1.5625 + 2.5t + t2 = 696.5 + 1.5625 Combine like terms: 696.5 + 1.5625 = 698.0625 1.5625 + 2.5t + t2 = 698.0625 Factor a perfect square on the left side: (t + 1.25)(t + 1.25) = 698.0625 Calculate the square root of the right side: 26.420872431 Break this problem into two subproblems by setting (t + 1.25) equal to 26.420872431 and -26.420872431.Subproblem 1
t + 1.25 = 26.420872431 Simplifying t + 1.25 = 26.420872431 Reorder the terms: 1.25 + t = 26.420872431 Solving 1.25 + t = 26.420872431 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + t = 26.420872431 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = 26.420872431 + -1.25 t = 26.420872431 + -1.25 Combine like terms: 26.420872431 + -1.25 = 25.170872431 t = 25.170872431 Simplifying t = 25.170872431Subproblem 2
t + 1.25 = -26.420872431 Simplifying t + 1.25 = -26.420872431 Reorder the terms: 1.25 + t = -26.420872431 Solving 1.25 + t = -26.420872431 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + t = -26.420872431 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = -26.420872431 + -1.25 t = -26.420872431 + -1.25 Combine like terms: -26.420872431 + -1.25 = -27.670872431 t = -27.670872431 Simplifying t = -27.670872431Solution
The solution to the problem is based on the solutions from the subproblems. t = {25.170872431, -27.670872431}
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