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Simplifying -5t2 + 23t + 10 = 5 Reorder the terms: 10 + 23t + -5t2 = 5 Solving 10 + 23t + -5t2 = 5 Solving for variable 't'. Reorder the terms: 10 + -5 + 23t + -5t2 = 5 + -5 Combine like terms: 10 + -5 = 5 5 + 23t + -5t2 = 5 + -5 Combine like terms: 5 + -5 = 0 5 + 23t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -1 + -4.6t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -4.6t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + -4.6t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -4.6t + t2 = 0 + 1 -4.6t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 -4.6t + t2 = 1 The t term is -4.6t. Take half its coefficient (-2.3). Square it (5.29) and add it to both sides. Add '5.29' to each side of the equation. -4.6t + 5.29 + t2 = 1 + 5.29 Reorder the terms: 5.29 + -4.6t + t2 = 1 + 5.29 Combine like terms: 1 + 5.29 = 6.29 5.29 + -4.6t + t2 = 6.29 Factor a perfect square on the left side: (t + -2.3)(t + -2.3) = 6.29 Calculate the square root of the right side: 2.507987241 Break this problem into two subproblems by setting (t + -2.3) equal to 2.507987241 and -2.507987241.Subproblem 1
t + -2.3 = 2.507987241 Simplifying t + -2.3 = 2.507987241 Reorder the terms: -2.3 + t = 2.507987241 Solving -2.3 + t = 2.507987241 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.3' to each side of the equation. -2.3 + 2.3 + t = 2.507987241 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + t = 2.507987241 + 2.3 t = 2.507987241 + 2.3 Combine like terms: 2.507987241 + 2.3 = 4.807987241 t = 4.807987241 Simplifying t = 4.807987241Subproblem 2
t + -2.3 = -2.507987241 Simplifying t + -2.3 = -2.507987241 Reorder the terms: -2.3 + t = -2.507987241 Solving -2.3 + t = -2.507987241 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.3' to each side of the equation. -2.3 + 2.3 + t = -2.507987241 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + t = -2.507987241 + 2.3 t = -2.507987241 + 2.3 Combine like terms: -2.507987241 + 2.3 = -0.207987241 t = -0.207987241 Simplifying t = -0.207987241Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.807987241, -0.207987241}
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