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Simplifying (11t + -30)(5t + -21) = t Reorder the terms: (-30 + 11t)(5t + -21) = t Reorder the terms: (-30 + 11t)(-21 + 5t) = t Multiply (-30 + 11t) * (-21 + 5t) (-30(-21 + 5t) + 11t * (-21 + 5t)) = t ((-21 * -30 + 5t * -30) + 11t * (-21 + 5t)) = t ((630 + -150t) + 11t * (-21 + 5t)) = t (630 + -150t + (-21 * 11t + 5t * 11t)) = t (630 + -150t + (-231t + 55t2)) = t Combine like terms: -150t + -231t = -381t (630 + -381t + 55t2) = t Solving 630 + -381t + 55t2 = t Solving for variable 't'. Reorder the terms: 630 + -381t + -1t + 55t2 = t + -1t Combine like terms: -381t + -1t = -382t 630 + -382t + 55t2 = t + -1t Combine like terms: t + -1t = 0 630 + -382t + 55t2 = 0 Begin completing the square. Divide all terms by 55 the coefficient of the squared term: Divide each side by '55'. 11.45454545 + -6.945454545t + t2 = 0 Move the constant term to the right: Add '-11.45454545' to each side of the equation. 11.45454545 + -6.945454545t + -11.45454545 + t2 = 0 + -11.45454545 Reorder the terms: 11.45454545 + -11.45454545 + -6.945454545t + t2 = 0 + -11.45454545 Combine like terms: 11.45454545 + -11.45454545 = 0.00000000 0.00000000 + -6.945454545t + t2 = 0 + -11.45454545 -6.945454545t + t2 = 0 + -11.45454545 Combine like terms: 0 + -11.45454545 = -11.45454545 -6.945454545t + t2 = -11.45454545 The t term is -6.945454545t. Take half its coefficient (-3.472727273). Square it (12.05983471) and add it to both sides. Add '12.05983471' to each side of the equation. -6.945454545t + 12.05983471 + t2 = -11.45454545 + 12.05983471 Reorder the terms: 12.05983471 + -6.945454545t + t2 = -11.45454545 + 12.05983471 Combine like terms: -11.45454545 + 12.05983471 = 0.60528926 12.05983471 + -6.945454545t + t2 = 0.60528926 Factor a perfect square on the left side: (t + -3.472727273)(t + -3.472727273) = 0.60528926 Calculate the square root of the right side: 0.77800338 Break this problem into two subproblems by setting (t + -3.472727273) equal to 0.77800338 and -0.77800338.Subproblem 1
t + -3.472727273 = 0.77800338 Simplifying t + -3.472727273 = 0.77800338 Reorder the terms: -3.472727273 + t = 0.77800338 Solving -3.472727273 + t = 0.77800338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.472727273' to each side of the equation. -3.472727273 + 3.472727273 + t = 0.77800338 + 3.472727273 Combine like terms: -3.472727273 + 3.472727273 = 0.000000000 0.000000000 + t = 0.77800338 + 3.472727273 t = 0.77800338 + 3.472727273 Combine like terms: 0.77800338 + 3.472727273 = 4.250730653 t = 4.250730653 Simplifying t = 4.250730653Subproblem 2
t + -3.472727273 = -0.77800338 Simplifying t + -3.472727273 = -0.77800338 Reorder the terms: -3.472727273 + t = -0.77800338 Solving -3.472727273 + t = -0.77800338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.472727273' to each side of the equation. -3.472727273 + 3.472727273 + t = -0.77800338 + 3.472727273 Combine like terms: -3.472727273 + 3.472727273 = 0.000000000 0.000000000 + t = -0.77800338 + 3.472727273 t = -0.77800338 + 3.472727273 Combine like terms: -0.77800338 + 3.472727273 = 2.694723893 t = 2.694723893 Simplifying t = 2.694723893Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.250730653, 2.694723893}
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