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Simplifying (-5t2) + 5t + 45 = 0 Reorder the terms: 45 + 5t + (-5t2) = 0 Solving 45 + 5t + (-5t2) = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '5'. 5(9 + t + (-1t2)) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(9 + t + (-1t2))' equal to zero and attempt to solve: Simplifying 9 + t + (-1t2) = 0 Solving 9 + t + (-1t2) = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -9 + -1t + t2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -1t + 9 + t2 = 0 + 9 Reorder the terms: -9 + 9 + -1t + t2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -1t + t2 = 0 + 9 -1t + t2 = 0 + 9 Combine like terms: 0 + 9 = 9 -1t + t2 = 9 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. -1t + 0.25 + t2 = 9 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 9 + 0.25 Combine like terms: 9 + 0.25 = 9.25 0.25 + -1t + t2 = 9.25 Factor a perfect square on the left side: ((t) + 0.5)((t) + 0.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting ((t) + 0.5) equal to 3.041381265 and -3.041381265.Subproblem 1
(t) + 0.5 = 3.041381265 Simplifying (t) + 0.5 = 3.041381265 t + 0.5 = 3.041381265 Reorder the terms: 0.5 + t = 3.041381265 Solving 0.5 + t = 3.041381265 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 = 3.041381265 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 3.041381265 + -0.5 t = 3.041381265 + -0.5 Combine like terms: 3.041381265 + -0.5 = 2.541381265 t = 2.541381265 Simplifying t = 2.541381265Subproblem 2
(t) + 0.5 = -3.041381265 Simplifying (t) + 0.5 = -3.041381265 t + 0.5 = -3.041381265 Reorder the terms: 0.5 + t = -3.041381265 Solving 0.5 + t = -3.041381265 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 = -3.041381265 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -3.041381265 + -0.5 t = -3.041381265 + -0.5 Combine like terms: -3.041381265 + -0.5 = -3.541381265 t = -3.541381265 Simplifying t = -3.541381265Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.541381265, -3.541381265}Solution
t = {2.541381265, -3.541381265}
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