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Simplifying 5t2 + 14t + -23 = 0 Reorder the terms: -23 + 14t + 5t2 = 0 Solving -23 + 14t + 5t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -4.6 + 2.8t + t2 = 0 Move the constant term to the right: Add '4.6' to each side of the equation. -4.6 + 2.8t + 4.6 + t2 = 0 + 4.6 Reorder the terms: -4.6 + 4.6 + 2.8t + t2 = 0 + 4.6 Combine like terms: -4.6 + 4.6 = 0.0 0.0 + 2.8t + t2 = 0 + 4.6 2.8t + t2 = 0 + 4.6 Combine like terms: 0 + 4.6 = 4.6 2.8t + t2 = 4.6 The t term is 2.8t. Take half its coefficient (1.4). Square it (1.96) and add it to both sides. Add '1.96' to each side of the equation. 2.8t + 1.96 + t2 = 4.6 + 1.96 Reorder the terms: 1.96 + 2.8t + t2 = 4.6 + 1.96 Combine like terms: 4.6 + 1.96 = 6.56 1.96 + 2.8t + t2 = 6.56 Factor a perfect square on the left side: (t + 1.4)(t + 1.4) = 6.56 Calculate the square root of the right side: 2.561249695 Break this problem into two subproblems by setting (t + 1.4) equal to 2.561249695 and -2.561249695.Subproblem 1
t + 1.4 = 2.561249695 Simplifying t + 1.4 = 2.561249695 Reorder the terms: 1.4 + t = 2.561249695 Solving 1.4 + t = 2.561249695 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.4' to each side of the equation. 1.4 + -1.4 + t = 2.561249695 + -1.4 Combine like terms: 1.4 + -1.4 = 0.0 0.0 + t = 2.561249695 + -1.4 t = 2.561249695 + -1.4 Combine like terms: 2.561249695 + -1.4 = 1.161249695 t = 1.161249695 Simplifying t = 1.161249695Subproblem 2
t + 1.4 = -2.561249695 Simplifying t + 1.4 = -2.561249695 Reorder the terms: 1.4 + t = -2.561249695 Solving 1.4 + t = -2.561249695 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.4' to each side of the equation. 1.4 + -1.4 + t = -2.561249695 + -1.4 Combine like terms: 1.4 + -1.4 = 0.0 0.0 + t = -2.561249695 + -1.4 t = -2.561249695 + -1.4 Combine like terms: -2.561249695 + -1.4 = -3.961249695 t = -3.961249695 Simplifying t = -3.961249695Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.161249695, -3.961249695}
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