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Simplifying 4t2 + 8t + 4 = 19 Reorder the terms: 4 + 8t + 4t2 = 19 Solving 4 + 8t + 4t2 = 19 Solving for variable 't'. Reorder the terms: 4 + -19 + 8t + 4t2 = 19 + -19 Combine like terms: 4 + -19 = -15 -15 + 8t + 4t2 = 19 + -19 Combine like terms: 19 + -19 = 0 -15 + 8t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -3.75 + 2t + t2 = 0 Move the constant term to the right: Add '3.75' to each side of the equation. -3.75 + 2t + 3.75 + t2 = 0 + 3.75 Reorder the terms: -3.75 + 3.75 + 2t + t2 = 0 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + 2t + t2 = 0 + 3.75 2t + t2 = 0 + 3.75 Combine like terms: 0 + 3.75 = 3.75 2t + t2 = 3.75 The t term is 2t. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2t + 1 + t2 = 3.75 + 1 Reorder the terms: 1 + 2t + t2 = 3.75 + 1 Combine like terms: 3.75 + 1 = 4.75 1 + 2t + t2 = 4.75 Factor a perfect square on the left side: (t + 1)(t + 1) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (t + 1) equal to 2.179449472 and -2.179449472.Subproblem 1
t + 1 = 2.179449472 Simplifying t + 1 = 2.179449472 Reorder the terms: 1 + t = 2.179449472 Solving 1 + t = 2.179449472 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = 2.179449472 + -1 Combine like terms: 1 + -1 = 0 0 + t = 2.179449472 + -1 t = 2.179449472 + -1 Combine like terms: 2.179449472 + -1 = 1.179449472 t = 1.179449472 Simplifying t = 1.179449472Subproblem 2
t + 1 = -2.179449472 Simplifying t + 1 = -2.179449472 Reorder the terms: 1 + t = -2.179449472 Solving 1 + t = -2.179449472 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = -2.179449472 + -1 Combine like terms: 1 + -1 = 0 0 + t = -2.179449472 + -1 t = -2.179449472 + -1 Combine like terms: -2.179449472 + -1 = -3.179449472 t = -3.179449472 Simplifying t = -3.179449472Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.179449472, -3.179449472}
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