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Simplifying 19 = 2t2 + -3t Reorder the terms: 19 = -3t + 2t2 Solving 19 = -3t + 2t2 Solving for variable 't'. Reorder the terms: 19 + 3t + -2t2 = -3t + 3t + 2t2 + -2t2 Combine like terms: -3t + 3t = 0 19 + 3t + -2t2 = 0 + 2t2 + -2t2 19 + 3t + -2t2 = 2t2 + -2t2 Combine like terms: 2t2 + -2t2 = 0 19 + 3t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -9.5 + -1.5t + t2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + -1.5t + 9.5 + t2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + -1.5t + t2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + -1.5t + t2 = 0 + 9.5 -1.5t + t2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 -1.5t + t2 = 9.5 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 9.5 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 9.5 + 0.5625 Combine like terms: 9.5 + 0.5625 = 10.0625 0.5625 + -1.5t + t2 = 10.0625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 10.0625 Calculate the square root of the right side: 3.172144385 Break this problem into two subproblems by setting (t + -0.75) equal to 3.172144385 and -3.172144385.Subproblem 1
t + -0.75 = 3.172144385 Simplifying t + -0.75 = 3.172144385 Reorder the terms: -0.75 + t = 3.172144385 Solving -0.75 + t = 3.172144385 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 3.172144385 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 3.172144385 + 0.75 t = 3.172144385 + 0.75 Combine like terms: 3.172144385 + 0.75 = 3.922144385 t = 3.922144385 Simplifying t = 3.922144385Subproblem 2
t + -0.75 = -3.172144385 Simplifying t + -0.75 = -3.172144385 Reorder the terms: -0.75 + t = -3.172144385 Solving -0.75 + t = -3.172144385 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -3.172144385 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -3.172144385 + 0.75 t = -3.172144385 + 0.75 Combine like terms: -3.172144385 + 0.75 = -2.422144385 t = -2.422144385 Simplifying t = -2.422144385Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.922144385, -2.422144385}
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