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Simplifying 3t + 8t = t2 + -3t + -12 Combine like terms: 3t + 8t = 11t 11t = t2 + -3t + -12 Reorder the terms: 11t = -12 + -3t + t2 Solving 11t = -12 + -3t + t2 Solving for variable 't'. Reorder the terms: 12 + 11t + 3t + -1t2 = -12 + -3t + t2 + 12 + 3t + -1t2 Combine like terms: 11t + 3t = 14t 12 + 14t + -1t2 = -12 + -3t + t2 + 12 + 3t + -1t2 Reorder the terms: 12 + 14t + -1t2 = -12 + 12 + -3t + 3t + t2 + -1t2 Combine like terms: -12 + 12 = 0 12 + 14t + -1t2 = 0 + -3t + 3t + t2 + -1t2 12 + 14t + -1t2 = -3t + 3t + t2 + -1t2 Combine like terms: -3t + 3t = 0 12 + 14t + -1t2 = 0 + t2 + -1t2 12 + 14t + -1t2 = t2 + -1t2 Combine like terms: t2 + -1t2 = 0 12 + 14t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -12 + -14t + t2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + -14t + 12 + t2 = 0 + 12 Reorder the terms: -12 + 12 + -14t + t2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + -14t + t2 = 0 + 12 -14t + t2 = 0 + 12 Combine like terms: 0 + 12 = 12 -14t + t2 = 12 The t term is -14t. Take half its coefficient (-7). Square it (49) and add it to both sides. Add '49' to each side of the equation. -14t + 49 + t2 = 12 + 49 Reorder the terms: 49 + -14t + t2 = 12 + 49 Combine like terms: 12 + 49 = 61 49 + -14t + t2 = 61 Factor a perfect square on the left side: (t + -7)(t + -7) = 61 Calculate the square root of the right side: 7.810249676 Break this problem into two subproblems by setting (t + -7) equal to 7.810249676 and -7.810249676.Subproblem 1
t + -7 = 7.810249676 Simplifying t + -7 = 7.810249676 Reorder the terms: -7 + t = 7.810249676 Solving -7 + t = 7.810249676 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + t = 7.810249676 + 7 Combine like terms: -7 + 7 = 0 0 + t = 7.810249676 + 7 t = 7.810249676 + 7 Combine like terms: 7.810249676 + 7 = 14.810249676 t = 14.810249676 Simplifying t = 14.810249676Subproblem 2
t + -7 = -7.810249676 Simplifying t + -7 = -7.810249676 Reorder the terms: -7 + t = -7.810249676 Solving -7 + t = -7.810249676 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + t = -7.810249676 + 7 Combine like terms: -7 + 7 = 0 0 + t = -7.810249676 + 7 t = -7.810249676 + 7 Combine like terms: -7.810249676 + 7 = -0.810249676 t = -0.810249676 Simplifying t = -0.810249676Solution
The solution to the problem is based on the solutions from the subproblems. t = {14.810249676, -0.810249676}
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