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Simplifying -4t(t + -2) + 6t = 5t + -4 Reorder the terms: -4t(-2 + t) + 6t = 5t + -4 (-2 * -4t + t * -4t) + 6t = 5t + -4 (8t + -4t2) + 6t = 5t + -4 Reorder the terms: 8t + 6t + -4t2 = 5t + -4 Combine like terms: 8t + 6t = 14t 14t + -4t2 = 5t + -4 Reorder the terms: 14t + -4t2 = -4 + 5t Solving 14t + -4t2 = -4 + 5t Solving for variable 't'. Reorder the terms: 4 + 14t + -5t + -4t2 = -4 + 5t + 4 + -5t Combine like terms: 14t + -5t = 9t 4 + 9t + -4t2 = -4 + 5t + 4 + -5t Reorder the terms: 4 + 9t + -4t2 = -4 + 4 + 5t + -5t Combine like terms: -4 + 4 = 0 4 + 9t + -4t2 = 0 + 5t + -5t 4 + 9t + -4t2 = 5t + -5t Combine like terms: 5t + -5t = 0 4 + 9t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1 + -2.25t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -2.25t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + -2.25t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2.25t + t2 = 0 + 1 -2.25t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2.25t + t2 = 1 The t term is -2.25t. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25t + 1.265625 + t2 = 1 + 1.265625 Reorder the terms: 1.265625 + -2.25t + t2 = 1 + 1.265625 Combine like terms: 1 + 1.265625 = 2.265625 1.265625 + -2.25t + t2 = 2.265625 Factor a perfect square on the left side: (t + -1.125)(t + -1.125) = 2.265625 Calculate the square root of the right side: 1.505199322 Break this problem into two subproblems by setting (t + -1.125) equal to 1.505199322 and -1.505199322.Subproblem 1
t + -1.125 = 1.505199322 Simplifying t + -1.125 = 1.505199322 Reorder the terms: -1.125 + t = 1.505199322 Solving -1.125 + t = 1.505199322 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + t = 1.505199322 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = 1.505199322 + 1.125 t = 1.505199322 + 1.125 Combine like terms: 1.505199322 + 1.125 = 2.630199322 t = 2.630199322 Simplifying t = 2.630199322Subproblem 2
t + -1.125 = -1.505199322 Simplifying t + -1.125 = -1.505199322 Reorder the terms: -1.125 + t = -1.505199322 Solving -1.125 + t = -1.505199322 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + t = -1.505199322 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = -1.505199322 + 1.125 t = -1.505199322 + 1.125 Combine like terms: -1.505199322 + 1.125 = -0.380199322 t = -0.380199322 Simplifying t = -0.380199322Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.630199322, -0.380199322}
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