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Simplifying 75t2 + 75t + -90 = 0 Reorder the terms: -90 + 75t + 75t2 = 0 Solving -90 + 75t + 75t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '15'. 15(-6 + 5t + 5t2) = 0 Ignore the factor 15.Subproblem 1
Set the factor '(-6 + 5t + 5t2)' equal to zero and attempt to solve: Simplifying -6 + 5t + 5t2 = 0 Solving -6 + 5t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.2 + t + t2 = 0 Move the constant term to the right: Add '1.2' to each side of the equation. -1.2 + t + 1.2 + t2 = 0 + 1.2 Reorder the terms: -1.2 + 1.2 + t + t2 = 0 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + t + t2 = 0 + 1.2 t + t2 = 0 + 1.2 Combine like terms: 0 + 1.2 = 1.2 t + t2 = 1.2 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + t2 = 1.2 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = 1.2 + 0.25 Combine like terms: 1.2 + 0.25 = 1.45 1.25 + t2 = 1.45 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 1.45 Calculate the square root of the right side: 1.204159458 Break this problem into two subproblems by setting (t + 0.5) equal to 1.204159458 and -1.204159458.Subproblem 1
t + 0.5 = 1.204159458 Simplifying t + 0.5 = 1.204159458 Reorder the terms: 0.5 + t = 1.204159458 Solving 0.5 + t = 1.204159458 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 1.204159458 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 1.204159458 + -0.5 t = 1.204159458 + -0.5 Combine like terms: 1.204159458 + -0.5 = 0.704159458 t = 0.704159458 Simplifying t = 0.704159458Subproblem 2
t + 0.5 = -1.204159458 Simplifying t + 0.5 = -1.204159458 Reorder the terms: 0.5 + t = -1.204159458 Solving 0.5 + t = -1.204159458 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -1.204159458 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -1.204159458 + -0.5 t = -1.204159458 + -0.5 Combine like terms: -1.204159458 + -0.5 = -1.704159458 t = -1.704159458 Simplifying t = -1.704159458Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.704159458, -1.704159458}Solution
t = {0.704159458, -1.704159458}
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