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Simplifying 30 = 6t2 + 6t Reorder the terms: 30 = 6t + 6t2 Solving 30 = 6t + 6t2 Solving for variable 't'. Reorder the terms: 30 + -6t + -6t2 = 6t + -6t + 6t2 + -6t2 Combine like terms: 6t + -6t = 0 30 + -6t + -6t2 = 0 + 6t2 + -6t2 30 + -6t + -6t2 = 6t2 + -6t2 Combine like terms: 6t2 + -6t2 = 0 30 + -6t + -6t2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(5 + -1t + -1t2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(5 + -1t + -1t2)' equal to zero and attempt to solve: Simplifying 5 + -1t + -1t2 = 0 Solving 5 + -1t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -5 + t + t2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + t + t2 = 0 + 5 t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 t + t2 = 5 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 = 5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = 5 + 0.25 Combine like terms: 5 + 0.25 = 5.25 1.25 + t2 = 5.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (t + 0.5) equal to 2.291287847 and -2.291287847.Subproblem 1
t + 0.5 = 2.291287847 Simplifying t + 0.5 = 2.291287847 Reorder the terms: 0.5 + t = 2.291287847 Solving 0.5 + t = 2.291287847 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 = 2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 2.291287847 + -0.5 t = 2.291287847 + -0.5 Combine like terms: 2.291287847 + -0.5 = 1.791287847 t = 1.791287847 Simplifying t = 1.791287847Subproblem 2
t + 0.5 = -2.291287847 Simplifying t + 0.5 = -2.291287847 Reorder the terms: 0.5 + t = -2.291287847 Solving 0.5 + t = -2.291287847 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 = -2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -2.291287847 + -0.5 t = -2.291287847 + -0.5 Combine like terms: -2.291287847 + -0.5 = -2.791287847 t = -2.791287847 Simplifying t = -2.791287847Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.791287847, -2.791287847}Solution
t = {1.791287847, -2.791287847}
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