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Simplifying 312 = 128t + -1t2 Solving 312 = 128t + -1t2 Solving for variable 't'. Reorder the terms: 312 + -128t + t2 = 128t + -128t + -1t2 + t2 Combine like terms: 128t + -128t = 0 312 + -128t + t2 = 0 + -1t2 + t2 312 + -128t + t2 = -1t2 + t2 Combine like terms: -1t2 + t2 = 0 312 + -128t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-312' to each side of the equation. 312 + -128t + -312 + t2 = 0 + -312 Reorder the terms: 312 + -312 + -128t + t2 = 0 + -312 Combine like terms: 312 + -312 = 0 0 + -128t + t2 = 0 + -312 -128t + t2 = 0 + -312 Combine like terms: 0 + -312 = -312 -128t + t2 = -312 The t term is -128t. Take half its coefficient (-64). Square it (4096) and add it to both sides. Add '4096' to each side of the equation. -128t + 4096 + t2 = -312 + 4096 Reorder the terms: 4096 + -128t + t2 = -312 + 4096 Combine like terms: -312 + 4096 = 3784 4096 + -128t + t2 = 3784 Factor a perfect square on the left side: (t + -64)(t + -64) = 3784 Calculate the square root of the right side: 61.514225997 Break this problem into two subproblems by setting (t + -64) equal to 61.514225997 and -61.514225997.Subproblem 1
t + -64 = 61.514225997 Simplifying t + -64 = 61.514225997 Reorder the terms: -64 + t = 61.514225997 Solving -64 + t = 61.514225997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '64' to each side of the equation. -64 + 64 + t = 61.514225997 + 64 Combine like terms: -64 + 64 = 0 0 + t = 61.514225997 + 64 t = 61.514225997 + 64 Combine like terms: 61.514225997 + 64 = 125.514225997 t = 125.514225997 Simplifying t = 125.514225997Subproblem 2
t + -64 = -61.514225997 Simplifying t + -64 = -61.514225997 Reorder the terms: -64 + t = -61.514225997 Solving -64 + t = -61.514225997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '64' to each side of the equation. -64 + 64 + t = -61.514225997 + 64 Combine like terms: -64 + 64 = 0 0 + t = -61.514225997 + 64 t = -61.514225997 + 64 Combine like terms: -61.514225997 + 64 = 2.485774003 t = 2.485774003 Simplifying t = 2.485774003Solution
The solution to the problem is based on the solutions from the subproblems. t = {125.514225997, 2.485774003}
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